Partial hyperbolicity and pseudo-Anosov dynamics
PPARTIAL HYPERBOLICITY AND PSEUDO-ANOSOVDYNAMICS
SERGIO R. FENLEY AND RAFAEL POTRIE
Abstract.
We show that if a hyperbolic 3-manifold admits a partially hyper-bolic diffeomorphism then it also admits an Anosov flow. Moreover, we give acomplete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifoldsinducing pseudo-Anosov dynamics in the base. This classification is given interms of the structure of their center (branching) foliations and the notion ofcollapsed Anosov flows. Introduction
A diffeomorphism f : M Ñ M of a closed 3-manifold is partially hyperbolic if its tangent bundle T M splits as a Df -invariant sum T M “ E s ‘ E c ‘ E u ofone-dimensional continuous subbundles and there exists (cid:96) ą v s , v c , v u are unit vectors in E s p x q , E c p x q and E u p x q respectively, then: } Df (cid:96) v s } ă min t , } Df (cid:96) v c }u and } Df (cid:96) v u } ą max t , } Df (cid:96) v c }u . This paper is concerned with the classification problem of partially hyperbolicdiffeomorphisms in dimension 3.It has become apparent that there is a strong link between partially hyperbolicdiffeomorphisms and Anosov flows in dimension 3, at least when the manifold is“sufficiently large”. This goes back at least to Pujals’ conjecture [BW] ´ whichroughly states that under certain very general conditions, the diffeomorphism isa variable time map of a topological Anosov flow. Recently new examples [BPP,BGP, BGHP] have been constructed which fail Pujals’ conjecture, for instancein Seifert manifolds. This has challenged our understanding of the topologicalstructure of these systems. This paper aims to solve the classification problem(as formulated in [BFP, Question 1]) completely for some particularly relevantclasses of manifolds and isotopy classes of maps.The first result concerns the problem of finding topological obstructions, or inother words to determine exactly when a manifold admits a partially hyperbolicdiffeomorphism. This problem is well understood when the manifold has (virtu-ally solvable) fundamental group [HP], or when it is Seifert fibered under someassumptions [HaPS]. It is always possible to construct a partially hyperbolic dif-feomorphism from an Anosov flow (in any manifold) by taking its time-one map.However it is expected that partially hyperbolic diffeomorphisms are much moreabundant amongst manifolds than Anosov flows. For example the 3 torus (cid:84) Mathematics Subject Classification.
Key words and phrases.
Partial hyperbolicity, 3-manifold topology, foliations, classification.S.F was partially supported by Simons Foundation grants numbered 280429 and 637554. R.P. was partially supported by CSIC 618, FCE-1-2017-1-135352. Part of this work was completedwhile R.P. was a Von Neumann fellow at the Institute for Advanced Study, funded by MinervaResearch Fundation Membership Fund and NSF DMS-1638352. We would like to thank the IASfor the great working environment. a r X i v : . [ m a t h . D S ] F e b S. FENLEY AND R. POTRIE or nil manifolds admit partially hyperbolic diffeomorphisms, but do not admitAnosov flows. This is because in these cases the fundamental group does nothave exponential growth, a necessary condition for the existence of an Anosovflow, by work of Margulis [Mar].One big focus of this paper is the case of hyperbolic 3-manifolds, that is, thosehomeomorphic to a quotient of (cid:72) by a cocompact group of isometries. These3-manifolds are by far the most abundant in the class of closed, irreducible 3-manifolds with infinite fundamental group, by the famous work of Thurston andPerelman.Our first result is the following: Theorem A.
Let M be a closed hyperbolic 3-manifold admitting a partially hy-perbolic diffeomorphism. Then, M admits an Anosov flow. One consequence of Theorem A is that it gives a complete set of obstructionsup to the problem of determining which hyperbolic 3-manifolds admit Anosovflows. It is unknown which hyperbolic 3-manifolds admit Anosov flows, thoughsome obstructions and examples are known [Ca ] (see § , BFFP ] which deals, amongst many other things, with generalpartially hyperbolic diffeomorphisms in hyperbolic 3-manifolds; and in addition[BFFP ] which considers partially hyperbolic diffeomorphisms in certain isotopyclasses of diffeomorphisms of Seifert manifolds. Our presentation aims to give aunified framework for both hyperbolic and Seifert 3-manifolds, which can also beapplied in other situations. We also develop a new tool to understand dynam-ical systems which preserve (branching) (cid:82) -covered, uniform foliations by usingcoarse dynamics. We call this tool a pseudo-Anosov pair . This is a pair of dif-feomorphisms which preserve branching foliations, and which lift to maps in theuniversal cover satisfying certain coarse geometric properties.Motivated by the present work, in [BFP] we propose a notion of collapsedAnosov flows . This relates partially hyperbolic diffeomorphisms with Anosovflows and their self-orbit equivalences. This covers all known examples of par-tially hyperbolic diffeomorphisms in manifolds with non solvable fundamentalgroup [BFP]. The concept of collapsed Anosov flow generalizes the notion ofleaf conjugacy to the case when f may not be dynamically coherent . In otherwords f may not admit f -invariant foliations tangent to the center stable andcenter unstable bundles. The dynamically incoherent situation is unavoidableand very common, as shown for example in Seifert manifolds [BGHP] (see also[BFFP , BFFP , Pot]). These works show that one needs new tools and modelsto attack a complete classification of partially hyperbolic diffeomorphisms. Ourresults provide a complete topological classification of partially hyperbolic diffeo-morphisms up to the center direction in both hyperbolic 3-manifolds and someisotopy classes of partially hyperbolic diffeomorphisms in Seifert manifolds.1.1. Collapsed Anosov flows.
Recall that an
Anosov flow is a C -flow φ t : M Ñ M whose time one map is a partially hyperbolic diffeomorphism (see § self orbit equivalence of φ t ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 3 is a homeomorphism β : M Ñ M such that sends orbits of φ t to orbits of φ t preserving orientation. These notions are discussed in more detail in [BFP] (see § Definition 1.1 (Collapsed Anosov flow) . A partially hyperbolic diffeomorphism f : M Ñ M is a collapsed Anosov flow if there is a topological Anosov flow φ t : M Ñ M , a self orbit equivalence β : M Ñ M of φ t and a continuous map h : M Ñ M homotopic to the identity such that:(i) h maps orbits of the flow injectively onto C curves tangent to the centerdirection E c of f ,(ii) one has that f ˝ h “ h ˝ β .In [BFP] we studied this definition and many of its possible variants. Weshowed that there are many examples of partially hyperbolic diffeomorphismsin 3-manifolds verifying this definition. We also studied different equivalent for-mulations and conditions that ensure that a partially hyperbolic diffeomorphismverifies this property.1.2. Statements.
In this paper we will show the following result:
Theorem B.
Let f : M Ñ M be a partially hyperbolic diffeomorphism on ahyperbolic 3-manifold. Then, it is a collapsed Anosov flow. Theorem B builds on [BFFP ] where a dichotomy is given for partially hy-perbolic diffeomorphisms in a closed hyperbolic 3-manifold: an iterate of f iseither a discretized Anosov flow (cf. § double translation (cf. § f “ φ t p x q p x q where φ t is a topologicalAnosov flow. This is a generalization of the time one map of an Anosov flow. Inparticular, discretized Anosov flows are collapsed Anosov flows. In this paper wefurther study the other possibility. In other words we study the double transla-tion case to obtain that in this case it must also be a collapsed Anosov flow. Weremark that in the double translation case f cannot be dynamically coherent andthe topological Anosov flow we will construct is (cid:82) -covered [BFFP ].Any topological Anosov flow in an atoroidal manifold is transitive [Mos]. Shan-non [Sha] recently announced that any transitive topological Anosov flow is or-bitally equivalent to an Anosov flow. This implies that Theorem A is a directconsequence of Theorem B. Two flows are orbitally equivalent if there is a home-omorphim sending orbits of the first into orbits of the second and preserving flowdirection.It is important to emphasize here that among the difficulties in showing The-orem B is the need to show that one does not need to take a finite cover or aniterate of f to obtain the result. In order to deal with this problem, we need toobtain strong uniqueness properties of the curves tangent to the center direction.More specifically, the results are obtained using branching foliations. The funda-mental results of Burago and Ivanov [BI] show that these exist for an iterate of f lifted to a finite cover. The finite cover has to do with orienting the bundlesof the partially hyperbolic diffeomorphism and so that Df preserves orientation.So a priori we obtain our results on a finite cover of M and for an iterate of With the results in this article we also prove that the map h maps weak stable and weakunstable leaves of φ t into C surfaces tangent respectively to E cs and E cu . See the discussionin [BFP]. For the purposes of this introduction, this definition will be ok. S. FENLEY AND R. POTRIE f . Uniqueness of branching foliations allows us to go back to M to obtain theannounced result in M and for f itself.In [BFFP , Theorem A] we got a complete classification of partially hyperbolicdiffeomorphisms on Seifert manifolds homotopic to the identity. Further resultsin this class of manifolds were obtained in [BFFP ] and will be used here. Herewe treat new isotopy classes: Theorem C.
Let f : M Ñ M be a partially hyperbolic diffeomorphism on aSeifert manifold with hyperbolic base, so that f acts as a pseudo-Anosov in thebase, then, it is a collapsed Anosov flow. Some results work only assuming that the action in the base has at least onepseudo-Anosov component and these will be stated in § [BGHP, § §
10. These results provide a complete classification of partiallyhyperbolic diffeomorphisms in these isotopy classes.One interesting point is that Theorems B and C admit mostly a unified proof,and we made an effort in presenting the unified point of view. We also show thatthis unified approach is helpful to study partially hyperbolic diffeomorphisms inother 3-dimensional manifolds. The difference between the proofs of TheoremsB and C has to do with how we show that certain general assumptions are met.Under these assumptions, we will get some even stronger results about classifica-tion (see §
11 and § Remark . In fact, for both Theorem B and C we obtain a stronger prop-erty which we called strong collapsed Anosov flow in [BFP]. See § Idea of proof.
Let us discus a bit the main difficulties we need to addressand the new tools we develop to take care of them.We focus on Theorem A and forget at first about the orientability issues men-tioned above which involve a different kind of problems that are discussed in § §
10. In other words we assume the necessary orientability conditions. As ex-plained, from the work of [BFFP ] we can reduce to the double translation case:the partially hyperbolic diffeomorphism preserves two (branching) foliations in M that are uniform and (cid:82) -covered. The lift r f of the homotopy of f to the iden-tity translates both such foliations. However, in principle we know nothing abouthow these foliations intersect, nor how they look at a big scale. The main drivinggoal we pursued in this project was the attempt to obtain geometric properties In [BFP, §
10] it is shown that not only small perturbations have this property but allpartially hyperbolic diffeomorphisms that can be connected to the examples by a path of partiallyhyperbolic diffeomorphisms. It is unknown if the space of partially hyperbolic diffeomorphismsis connected in this isotopy class.
ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 5 of the intersection of these (branching) foliations by showing that the intersectedleaves are quasigeodesics in the leaves of each branching foliation when lifted tothe universal cover.This strong geometric property is proved in steps. We consider the centerfoliations in (say) center stable leaves lifted to the universal cover. Each suchcenter stable leaf is Gromov hyperbolic and is compactified to a closed disk withan ideal circle. We first show that for each ray in a center stable leaf L , the rayaccumulates on a single point in the ideal circle of L in § c in L , the ideal points of the two rays of c are distinct idealpoints of L . Finally we show that for any center stable leaf L in the universalcover, then the leaf space of the center foliation in L is Hausdorff in §
6. Togetherthese properties then imply that the centers are uniform quasigeodesics in thecenter stable leaves as proved in § § pseudo-Anosov pairs . In our setting these are choices of a lift of f to theuniversal cover together with a deck transformations that allow us to find somecoarse dynamical properties that interact in a very precise way with foliationpreserving diffeomorphisms. However, while such configurations alone allow toobtain many of the properties we need, they are not enough to deal with all thepotential configurations that obstruct the quasigeodesic behavior of the centers,and we need to use what we call full pseudo-Anosov pairs which have the propertythat their conjugates can be made to interact in an appropriately strong formwith the entire universal circle of the foliation.We also introduce the notion of super attracting fixed points for actions on theuniversal circle in Subsection 2.5. This notion was first introduced in the settingof lifts of homeomorphisms of closed surfaces in [BFFP ]. Here we generalizethis notion to the case of (cid:82) -covered, uniform foliations with Gromov hyperbolicleaves. This notion plays a fundamental role in the definition of pseudo-Anosovpairs and the properties that can be proved from pseudo-Anosov pairs. We expectthat it will also be useful in other contexts.Finally we consider orientability issues: the results above use branching folia-tions, which assume taking an iterate of f lifted to a finite cover of M . We thenprove invariance of these branching foliations under deck transformations of thefinite cover. To obtain this we strongly use the quasigeodesic behavior we provein the cover. The result is that the branching foliations descend to M and aniterate of f satisfies all the orientability conditions. Then using additional resultsof Burago and Ivanov [BI] we approximate the center stable and center unstablefoliations by foliations F and G which intersect along a one dimensional orientedfoliation, generating a flow. We show that this flow is expansive, generating atopological Anosov flow. By the result of Shannon [Sha] the flow is orbitallyequivalent to an Anosov flow finishing the proof.We note that additional arguments are needed to obtain the proof of TheoremB.1.4. Context and comments.
The problem of the topological classification ofAnosov flows in 3-manifolds goes back to the seminal work of Margulis [Mar] andPlante-Thurston [PT] showing that a 3-manifold admitting an Anosov flow musthave exponential growth of fundamental group. It is noteworthy that when theseresults appeared, the only known examples were orbitally equivalent to geodesicflows in the unit tangent bundle of negatively curved manifolds and suspension oftoral automorphisms. Since then, a myriad of new very different examples haveappeared starting with the ones constructed by Franks-Williams [FrWi] and those
S. FENLEY AND R. POTRIE by Handel-Thurston [HaTh] and Goodman [Go] with somewhat different meth-ods (we refer the reader to the introduction of [BBY] for a list of known examplesand constructions). This was just the beginning. In hyperbolic manifolds startingwith the fundamental work of Goodman [Go], new examples have continued to ap-pear until very recently (see for instance [Fen , FH, BM]). Questions about which3-manifolds support Anosov flows and how many orbitally inequivalent ones suchmanifolds admit are still abundant. There has been considerable progress on theclassification of Anosov flows in manifolds with non-trivial JSJ decomposition,we mention the recent work of Barbot and the first author in particular whichgives a rather complete classification of what they call totally periodic Anosovflows in graph manifolds as well as other classes, see [BaFe , BaFe , BaFe ]. Werefer the reader to [Ba , Bart] for surveys about Anosov flows in dimension 3.The case of hyperbolic 3-manifolds is certainly the most mysterious. There aresome known obstructions for hyperbolic manifolds to admit Anosov flows, andseveral constructions of such flows. Recently, some hyperbolic 3-manifolds havebeen shown to admit an arbitrary large number of orbitally inequivalent Anosovflows [BM]. All these results make our results somewhat more interesting, since itimplies that we cannot compare our systems with some model (Anosov) systemsin the manifold, as is the case for example in solvable manifolds. We point outin particular that the known topological obstructions to admit Anosov flows inhyperbolic 3-manifolds are very sensitive to taking finite covers (see [RRS, CD]).Hence it is very important for us to obtain the results in Theorem A without needto take finite covers (which introduces a big challenge, since our starting point isthe existence of branching foliations from [BI] which requires some orientablityassumptions).Let us first comment on our Theorems A and B. The first important thing topoint out is that they both rely heavily on our previous work with Barthelm´eand Frankel [BFFP ]: In that paper we showed that a partially hyperbolic dif-feomorphism of a hyperbolic 3-manifold (up to iterate so that it is homotopic tothe identity) is either a discretized Anosov flow or up to finite cover admits twotransverse taut (branching) foliations which were translated by the lift of the dy-namics to the universal cover. We point out that the existence of two transversetaut foliations (even with all possible orientability assumptions) in a hyperbolicmanifold does not imply the existence of an Anosov flow, at least not one relatedto those foliations (see [BBP]). Here, we analyze the second case, and describe itcompletely. A posteriori this leads to the existence of an Anosov flow in M .We mention that the proof of Theorem B is very similar to the proof of TheoremC and we present it in a way that the only difference is in how one shows thatcertain conditions are met, that we do at the very end. On the other hand,Theorem C is mostly self contained, since in the isotopy class under analysis, weonly need to deal with that case. (The analogy would be that the discretizedAnosov flow case is when f is homotopic to the identity in a Seifert manifold,which is the case we treated in [BFFP ].)Our results fit well in the program of classification of partially hyperbolicdiffeomorphisms in dimension 3 and have motivated the definition of collapsedAnosov flows which we believe may play an important role in this program. Werefer the reader to [Pot, HP] for recent surveys on the classification of partiallyhyperbolic diffeomorphisms in dimension 3. In [BFP], with Barthelm´e, we havedeveloped the notion of a collapsed Anosov flow that was suggested by this work.1.5. Organization of the paper.
After giving some preliminaries in § ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 7
Subsection 2.5. In § , §
8] and [BFFP , §
11] as well as [BFFP ], in particular using the notion of super attracting fixedpoints, and which also works in other settings. Part of section § , BFFP , BFFP ] to a more general setting.In § § §
13. This servestwo purposes, on the one hand it allows to obtain both results almost simultane-ously; on the other hand, it also intends to express precisely what properties weuse and where and allows to follow the arguments without prior knowledge onfine properties of hyperbolic 3-manifolds (the properties we will use only appearin § § § § §
12 (Theorems A and B are proved in §
11 and Theorem C is proved in § § § § § Preliminaries and discussions on some notions
In this paper M denotes a closed aspherical 3-manifold, and π : Ă M Ñ M the universal covering map. We will assume that the manifold does not have(virtually) solvable fundamental group. This allows to simplify some statements,and the case of (virtually) solvable fundamental group for partially hyperbolicdiffeomorphisms is already well understood (see [HP]). In some sections at theend of the paper we will restrict further to M being either Seifert or hyperbolic.Our results and statements will be independent of the chosen Riemannianmetric, but we will fix one first for which the definition of partial hyperbolicity isgiven, and later we will change the metric so that the leaves of the (branching)foliations are negatively curved: this only changes definitions by bounded factors.In this section we introduce some preliminaries and fix notations which willbe used later and relate with the objects introduced in the previous section.The reader familiar with [BFFP , BFP] can safely skip this section, except forSubsection 2.5 where the notion of super attracting fixed point in the universalcircle is introduced.2.1. Branching foliations.
We will give a brief account on what we need aboutbranching foliations introduced in [BI] in our context. For a more detailed accountwe refer the reader to [BFFP , §
3] or [BFP].Our definition will be a bit more restrictive (what we will define would be a
Reebless branching foliation ) which is more than enough for our purposes andmakes the definition easier to give.A branching ( -dimensional) foliation on a closed 3-manifold M is a collectionof immersed surfaces F tangent to a 2-dimensional continuous distribution E of T M such that if we consider r F the lift of the collection to Ă M we have the followingproperties:(i) Every leaf L P r F is a properly embedded plane separating Ă M into twoopen connected components L ‘ and L a depending on a fixed transverseorientation to E lifted to Ă M . Denote L ` “ L Y L ‘ and L ´ “ L Y L a .(ii) Every point x P Ă M belongs to at least one leaf L P r F . S. FENLEY AND R. POTRIE (iii) For every two leaves
L, F P r F we have that either F Ă L ` or F Ă L ´ .This is the no topological crossings condition.(iv) If x n Ñ x and L n P r F so that x n P L n . Then every limit of L n in thecompact-open topology belongs to r F .We will add an additional condition in the case that the distribution E istransversely oriented, which is that every diffeomorphism preserving E and itstransverse orientation preserves the branching foliation in the sense that the imageunder f of a leaf of F is a leaf of F .A branching foliation is well approximated by foliations if for every ε ą F ε tangent to a bundle E ε and continuous maps h ε : M Ñ M so that: ‚ The angle between E ε and E is smaller than ε . ‚ The map h ε is ε - C -close to the identity (in particular, it is homotopic tothe identity and therefore surjective) sending leaves of F ε to leaves of F . ‚ For every L P F there is a unique leaf L ε P F ε so that h ε : L ε Ñ L isa local C -diffeomorphism so that 1 ´ ε ă } Dh ´ ε } ´ ď } Dh ε } ă ` ε (therefore, in Ă M it lifts to a diffeomorphism).Note that when a branching foliation is well approximated by foliations we candefine a leaf space L F by identifying with the leaf space of some of the approxi-mating foliations L F ε “ Ă M { Ă F ε . This uses the uniqueness result in the third itemabove, so there is a bijection between the leaf spaces of r F and r F ε .We will use the following result (the uniqueness statement for the approximat-ing foliation is explained in [BFP, Appendix A]): Theorem 2.1 ([BI]) . Let f : M Ñ M be a partially hyperbolic diffeomorphismof a closed 3-dimensional manifold so that the bundles E s , E c , E u are orientedand Df preserves their orientation. Then, there exist branching foliations W cs and W cu tangent to E cs and E cu respectively. These branching foliations are wellapproximated by foliations. We will denote by L cs and L cu the leaf spaces respectively of the lifts Ą W cs , Ą W cu of the branching foliations. In our setting, we will be considering a special classof foliations where leaf spaces are easier to define. See [BFFP , § Ą W cs and Ą W cu gives rise to a one-dimensional branching foliation Ă W c which also has a well defined leaf space (see[BFP, § one dimensional branching foliation T which subfoliates a foli-ation F we mean a collection of C -curves such that in the universal cover, forevery L P r F the curves of r T contained in L have the same properties ( i )-( iv ) defin-ing two dimensional branching foliations (of course, in ( i ) one needs to changeproperly embedded plane to properly embedded line).2.2. (cid:82) -covered foliations and hyperbolic metrics. A celebrated result ofCandel [Can] states that under quite general conditions, given a foliation in a3-manifold, there is a metric on M that makes every leaf a hyperbolic surface.In particular, it follows from [FP, Theorem 5.1] that this is the case for everyminimal foliation in manifolds with exponential growth of π p M q as we will con-sider here. Here is whay: If there are no holonomy invariant transverse measures,then this result follows directly from Candel’s theorem [Can]. If there is a holo-nomy invariant transverse measure, then this result follows from [FP, Theorem5.1]. Since we will be mainly concerned with minimal (cid:82) -covered foliations we will ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 9 consider such a metric for some approximating foliations F cs , F cu to Ą W cs , Ą W cu and this will induce a (coarsely) negatively curved metric on each leaf of bothbranching foliations, see the next subsection.We will say that the branching foliation F is (cid:82) - covered if for every pair ofleaves L, F P r F we have that either L ` Ă F ` or F ` Ă L ` . This allows toinduce an order between leaves and therefore it is equivalent to having that theapproximating foliation is (cid:82) -covered, that is, the leaf space L F is homeomorphicto (cid:82) . Compare with [BFFP , § § uniform and (cid:82) -covered branching fo-liations. A branching foliation F is uniform if given two leaves L, F P r F theHausdorff distance between them in Ă M is finite. It follows from [FP , Theorem1.1] (see also [FP , § (cid:82) -covered (here weuse the hypothesis that the approximating foliations are Reebless). We will saythat a branching foliation is by hyperbolic leaves if the metric on M makes allleaves uniformly Gromov hyperbolic (see [BFP, § A.3]).2.3.
Boundary at infinity and visual metric.
Let L be a complete simplyconnected surface which is Gromov hyperbolic. We can define S p L q its visual (orGromov) boundary as the set of geodesic rays on L identified by being equivalentif they are a finite Hausdorff distance from each other in L . An oriented bi-infinitegeodesic (cid:96) P L has therefore two endpoints (cid:96) ˘ corresponding to the positively andnegatively oriented rays of (cid:96) zt x u for some x P (cid:96) . This is clearly independent ofthe point x P (cid:96) .One can compactify L to ˆ L “ L Y S p L q making it homeomorphic to theclosed disk (see [BH, § III.H.3] or [Fen ]). Given a geodesic (cid:96) in L and a point ξ in S p L qzt (cid:96) ˘ u we can define an open set O (cid:96) p ξ q containing ξ P S p L q as theunion of the open interval of S p L q whose endpoints are (cid:96) ` and (cid:96) ´ and contains ξ and the connected component of L z (cid:96) containing completely a geodesic ray r representing ξ . Note that for every ξ zt (cid:96) ˘ u there are rays in L z (cid:96) representing it,and the definition of the open set O (cid:96) p ξ q is independent of this choice. The opensets O (cid:96) p ξ q with (cid:96) being geodesics in (cid:96) together with the open sets in L give atopology in ˆ L making it homeomorphic to a closed disk.For several reasons, we will choose a metric in M whose restriction to leavesof F is CAT p κ q for some κ ă p´ q . This property implies uniqueness of geodesic segments, rays, orfull geodesics given the endpoints or ideal points. In particular the CAT p´ q property implies that for any x in a leaf L there is exactly one such ray startingat x for every ξ P S p L q so one can identify S p L q with T x L . This defines, foreach x P L a visual metric on S p L q given by measuring intervals in S p L q bythe angle in T x L measured with the Riemannian metric on L . A very importantfact for us is the following: Remark . The visual metric in S p L q is well defined up to H¨older equivalence[BH, § III.H.3].These metrics in the leaves vary continuously with the points in M . Then S p L q is canonically identified with T x L . Also when x varies, the sets T x L x varycontinuously. It follows that the visual metrics in S p L q vary continuously withthe points. We refer the reader to [Th , Fen , Ca , Ca ] for more general statements.2.4. The universal circle.
In this subsection we describe what we need of theuniversal circle of the foliation which allows us to determine the dynamics atinfinity of a good pair. What is described here is done with much more detailand richer properties in [Th , Fen , Ca ]. Our construction is from scratch withonly the properties we will need, see also [FP , § F be a uniform (cid:82) -covered branching foliation on M by hyperbolic leavesand r F its lift to Ă M . For each L P r F we consider its visual boundary S p L q as in § A “ Ů L P r F S p L q which can be given a topology from the collection of T r F | τ where τ is a transversal compact segment to r F . Then if L n Ñ L in L “ Ă M { r F then S p L n q Ñ S p L q . With this topology A is an open annulus since it is acircle bundle over the leaf space L of r F .Recall that a quasi-isometry between two metric spaces p X, d X q and p Y, d Y q isa map q : X Ñ Y such that there exists C ą x, x P X wehave: C ´ d X p x, x q ´ C ď d Y p q p x q , q p x qq ď Cd X p x, x q ` C. We do not require q to be continuous, the constant C is called a quasi-isometryconstant for q . A quasigeodesic in p X, d X q is a quasi-isometric map from (cid:82) withits usual distance into X and a quasigeodesic ray is a quasigeodesic map from r , to X . The Morse Lemma (see eg. [BH, Theorem III.H.1.7]) states thatif L is a Gromov hyperbolic simply connected surface then for every C there issome K such that if r is a C -quasigeodesic (resp. C -quasigeodesic ray), thenthere exists a true geodesic (resp. geodesic ray) at Hausdorff distance less than K from r . This also holds with the same constants for quasigeodesic segmentsand the geodesic joining the endpoints. Proposition 2.3.
Let f : M Ñ M be a homeomorphism preserving F , it followsthat any lift ˆ f extends naturally to a homeomorphism of A that we still call ˆ f and preserves the fibers (i.e. it is a homeomorphism from S p L q to S p ˆ f p L qq . Note that, by taking f “ id this includes action by deck transformations in Ă M . Proof.
Since f is a homeomorphism of M which is compact, then any given liftˆ f induces quasi-isometries from L to ˆ f p L q for every L P r F so it maps geodesicrays into quasigeodesic rays. The Morse Lemma implies that these are boundeddistance away from a well defined geodesic ray up to bounded distance. Thisinduces a continuous map from S p L q to S p ˆ f p L qq and ˆ f ´ induces its inverse soit is a homeomorphism. (cid:3) It is many times useful to collate all circles in S p L q by constructing a universalcircle , introduced by Thurston. There are standard constructions, which in thesetting of (cid:82) -covered uniform foliations gets simplified [Th , §
5] (see [Fen , Ca ,Ca ] for more details and more general constructions). One can cover the manifold by finitely many sufficiently small foliations charts. A homeo-morphism verifies that the image of a plaque in a foliation chart can intersect only finitely many(uniformly bounded number of) foliation charts. Since plaques in the chart have size uniformlybounded from above and below, one deduces that ˆ f must be a uniform quasi-isometry betweenleaves of the foliation. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 11
To do this for (cid:82) -covered uniform foliations, it is important to construct anatural way to identify leaves of r F . Intuitively, one can think as if there is a flowΦ t in M which is transverse and regulating to F : this means that if one considerstwo leaves L , L P r F then the time it takes the flow Φ t to take a point of L to a leaf in L is bounded above by a constant only depending on L and L .This flow can be extended to A and gives a way to identify fibers. Such a flowexists for general transversely oriented, uniform (cid:82) -covered foliations [Fen ]. Toconstruct the identification between distinct circles at infinity, less is needed: Proposition 2.4.
There is a family t τ L,L : L Y S p L q Ñ L Y S p L qu for L, L P r F with the following properties: (i) the map τ L,L | L : L Ñ L is a quasi-isometry whose constant depends onlyon the Hausdorff distance between L and L , (ii) the map τ L,L | S p L q : S p L q Ñ S p L q is a homeomorphism, (iii) one has that τ L ,L | S p L q ˝ τ L,L | S p L q “ τ L,L | S p L q . This statement can be found in [Th , § , Corollary 5.3.16] or [Fen ,Proposition 3.4] and the proof works exactly the same for branching foliations .The quasi-isometries τ L,L : L ÞÑ L are coarsely well defined, in the sense thatgiven L, L there is a constant b which depends only on the Hausdorff distancebetween L, L so that for any x in L , then d Ă M p x, τ L,L p x qq ă b . For (cid:82) -coveredfoliations this implies that if τ L,L is another such map then d L p τ L,L p x q , τ L,L p x qq ă b , for a constant b that depends only on b . So τ L,L is coarsely defined.We can now define S univ , the universal circle of the foliation F , as A { „ wherewe identify the circles S p L q and S p L q via the maps τ L,L from the proposi-tion. It is important to remark that the universal circle depends on the foliation,and so, when several foliations are involved (as is the case of partially hyper-bolic diffeomorphisms) we will make an effort to make clear which circle we areconsidering. For any L in r F defineΘ L : S univ ÞÑ S p L q the map that associates to a point in S univ its representative in S p L q . Noticethat for any leaves L, E of r F thenΘ E “ τ L,E ˝ Θ L A useful property for us is that the following extension of Proposition 2.3 holds:
Proposition 2.5.
Let f : M Ñ M be a homeomorphism preserving an (cid:82) -covereduniform branching foliation F by hyperbolic leaves and ˆ f a lift to Ă M . Then, thereis a well defined action ˆ f of ˆ f on S univ given by ˆ f “ Θ ´ L ˝ τ ˆ f p L q ,L ˝ ˆ f ˝ Θ L : S univ Ñ S univ , where L is an arbitrary leaf of r F . In other words the map ˆ f isindependent of the choice of L .Proof. We sketch the proof, for more details see the proof of [Fen , Proposition3.4]. Let p in S univ . Choose an arbitrary leaf L of r F to start with. The point p in S univ is associated with a point q in S p L q , q “ Θ L p p q . Let r be a geodesic ray in L with ideal point q . Then since ˆ f is a quasi-isometry from L to ˆ f p L q , it follows Or can be deduced for them by using approximating foliations. that ˆ f p r q is a quasigeodesic ray and has a unique ideal point q in S p ˆ f p L qq . Anyother geodesic ray r in L with ideal point q in L , r is asymptotic to r , henceˆ f p r q is a finite distance from ˆ f p r q and defines the same ideal point in ˆ f p L q .Finally we need to show that the map is independent of the choice of L , thatis, that ˆ f ˝ τ L,E | S p L q “ τ ˆ f p L q , ˆ f p E q ˝ ˆ f | S p L q . for any leaf E of r F .For this, let E be another leaf of r F . The map τ L,E : L ÞÑ E is a quasi-isometryso that for any x in L , then d Ă M p x, τ L,E p x qq ă b for b which depends only on theHausdorff distance between L, E . It follows that τ L,E p r q is a quasigeodesic rayin E which is a bounded distance in Ă M from r . The quasigeodesic ray τ L,E p r q isalso a bounded distance in E from a geodesic ray in E (this bound only dependson the quasi-isometry constant of τ L,E ). Hence there is a geodesic ray r in E which is a bounded distance in Ă M from r . If q is the ideal point of r in E ,then by definition τ L,E p q q “ q . Taking the image of both r, r by ˆ f we obtainquasigeodesic rays ˆ f p r q , ˆ f p r q in ˆ f p L q , ˆ f p E q respectively, which are a boundeddistance from each other in Ă M . The ideal point of ˆ f p r q is ˆ f p q q . The idealpoint of the second is ˆ f ˝ τ L,E p q q . Since these quasigeodesic rays in ˆ f p L q , ˆ f p E q respectively are a bounded distance from each other in Ă M , they define the samepoint in the universal circle, in other words τ ˆ f p L q , ˆ f p E q p ˆ f p q qq “ ˆ f p τ L,E p q qq , which is exactly what we wanted to prove. (cid:3) Visual metrics on the universal circle S univ . In the previous sectionwe described visual metrics in individual leaves of r F . It will be useful to havemetrics on S univ to talk about super attracting fixed points of homeomorphismsacting on S univ .Consider first a leaf L of r F . There is a well defined bijection Θ L : S univ Ñ S p L q . The ideal circle S p L q has visual metrics: given x in L there is a bijectionbetween the unit tangent vectors to T r F at x and the points in L . The angle metricin T x r F induces a metric on S p L q . When one changes the basepoint in L thevisual metric in S p L q changes by a H¨older homemorphism as noted in Remark2.2, see [BH, § III.H.3].A map g : p A, d q ÞÑ p
B, d q between metric spaces is quasisymmetric if ‚ g is an embedding, ‚ there is a homeomorphism η : r ,
8q ÞÑ r , so that if x, y, z are distinctpoints in A , then d p g p z q , g p x qq d p g p y q , g p x qq ď η ˆ d p z, x q d p y, x q ˙ See [Ha, Definition 2.1].When one changes from one leaf L to other leaf E , one needs to understandthe metric properties of the map τ L,E restricted to S p L q . It is the identificationassociated with the universal circle S univ . Recall from Proposition 2.4 that τ L,E is a quasi-isometry from L to E . Quasi-isometries between Gromov hyperbolicspaces induce quasi-symmetric homeomorphisms of the boundaries. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 13
One can obtain this from the proofs of [GH, Propositions 5.15 and 6.6]. How-ever they only explicitly talk about quasiconformal behavior, which in dimension1 does not provide much information. In [Ha, Theorem 3.1] there is an explicitproof that a C -quasi-isometry between Gromov hyperbolic spaces induces an idealmap g which is quasisymmetric with constants related to C , the quasi-isometryconstant. He proves that it is quasi-M¨obius (which we do not define here), whichimplies quasisymmetric. We note that the quasi-isometry constant C depends onthe Hausdorff distance between leaves but we do not have much control on it.However, some metric properties make sense. We can now introduce super at-tracting fixed points. Let d be a visual metric in S univ given say by identificationwith S p L q using a point in L . Definition 2.6.
Let f be a homeomorphism of S univ which fixes a point ξ in S univ . We say that ξ is a super attracting fixed point for f iflim x Ñ ξ d p h p x q , ξ q d p x, ξ q “ , where h is the expression of f using the identification of S univ with S p L q forsome leaf L of r F .Compare with the definition given in [BFFP , Appendix A] which is done in aspecial situation. The name is inspired in complex dynamics where super attract-ing points are those whose derivative vanishes. Here, even if we cannot define thederivative since maps are only continuous, the quasi-symmetric structure allows‘zero derivative’ as in Definition 2.6 to make sense as we will prove next. Lemma 2.7.
The property of ξ being a super attracting fixed point for a homeo-morphism f : S univ ÞÑ S univ is independent of the leaf L in r F .Proof. Let h be the expression of the homeomorphism f using the identificationΘ L : S univ Ñ S p L q . If S univ is identitifed with S p E q for another leaf E of r F ,then the visual metrics d in S univ coming from identification with S p L q and d from identification with S p E q are quasisymmetric using the map g which is τ L,E restricted to S p L q . Let η be the quasisymmetric function associated to g . Then d p g p h p x q , g p ξ qq d p g p x q , g p ξ qq ď η ˆ d p h p x q , ξ q d p x, ξ q ˙ . Here d is the metric in S univ coming from identification with S p E q . As η is ahomeomorphism with η p q “
0, it follows thatlim x Ñ ξ d p g p h p x q , g p ξ qq d p g p x q , g p ξ qq “ . Let z “ g p x q . Since g is a homeomorphism, then x limits to ξ if and only if z limits to g p ξ q . So we obtainlim z Ñ g p ξ q d p g p h p g ´ p z qqq , g p ξ qq d p z, g p ξ qq “ . But g ˝ h ˝ g ´ is the expression of f using E instead of L . This proves thelemma. (cid:3) Discretized and collapsed Anosov flows.
We refer the reader to thepaper [BFP] which discusses in detail these concepts, as well as equivalences,examples and properties. Here we just give some quick definitions and propertiesthat we will use to prove our results. Let M be a closed 3-manifold. A non-singular flow φ t : M Ñ M generated by a vector field X is said to be Anosov ifthere is a Dφ t -invariant splitting T M “ E s ‘ (cid:82) X ‘ E u and t ą v σ P E σ is a unit vector ( σ “ s, u ) then: } Dφ t v s } ă ă ă } Dφ t v u } . It is easy to show that a flow on M is Anosov if and only if its time 1 map (andtherefore its time t -map for every t ) is partially hyperbolic. We refer the readerto [Fen , Ba , Bart] for generalities on the topological properties of Anosov flows.We also have to consider the topological versions of these objects. A topologicalAnosov flow φ t : M Ñ M is an expansive flow tangent to a continuous vectorfield X which preserves two transverse foliations so that orbits of one of the foli-ations get contracted under forward flowing while orbits of the other foliation arecontracted by backward flowing. See [BFFP , Appendix G] for more discussions.It has recently been established by Shannon that transitive topologically Anosovflows are orbit equivalent to true Anosov flows [Sha].More generally, a pseudo-Anosov flow is a flow φ t : M Ñ M preserving twotransverse singular foliations which is locally modelled in a topological Anosovflow away from finitely many periodic orbits on which it has singularities of prongtype. See [Ca ] for more details. We note that every expansive flow is pseudo-Anosov [IM, Pat].In this paper we will be mostly interested in what is called (cid:82) - covered Anosovflows : that is, topological Anosov flows whose stable foliation F ws and unstablefoliations F wu lifted to Ă M are (cid:82) -covered. There are two important classes of (cid:82) -covered foliations: suspensions and skewed Anosov flows . A fundamental earlyresult on Anosov flows is the following: Theorem 2.8 ([Fen , Ba ]) . The orbit space of the lift r φ t of an arbitrary Anosovflow to Ă M is homeomorphic to (cid:82) . The flow is (cid:82) -covered if and only if oneof the foliations F ws or F wu is (cid:82) -covered. The foliations r F ws , r F wu induce one-dimensional foliations on (cid:82) and φ t is a suspension if and only if the foliationshave a global product structure. If φ t is (cid:82) -covered and not orbitally equivalentto a suspension, then φ t is skewed . Moreover, every (cid:82) -covered Anosov flow istransitive. Notice that the previous theorem is shown for topological Anosov flows, so,combined with [Sha] it says that if a topological Anosov flow is (cid:82) -covered, thenit is orbit equivalent to a true Anosov flow. Also, it follows from [Bru] thattopological Anosov flows in atoroidal 3-manifolds are always transitive.3.
Pseudo-Anosov good pairs
Let F be a Reebless branching foliation of a closed 3-manifold M which is (cid:82) -covered and uniform and by hyperbolic leaves. Denote by r F to the lift of F to Ă M . We choose a transverse orientation for r F and for L P r F we denote by L ` and L ´ the closed half spaces determined by L in Ă M .By the definition of branching foliation it follows that given L P r F , every leaf L P r F is contained in either L ` or L ´ (and if it is contained in both, it must be L ). We will denote by S univ the universal circle of F (cf. § ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 15
Good pairs.
We will be interested in certain lifts of maps that preserve thefoliation F . Definition 3.1.
Given f, g : M Ñ M diffeomorphisms of M preserving F , a pair p ˆ f , ˆ g q where ˆ f , ˆ g : Ă M Ñ Ă M are lifts of f, g is called a good pair if they commute,neither fixes a leaf of r F and one of them acts as the identity on S univ .Notice that this implies that both ˆ f and ˆ g act as a translation on the leaf space L F – (cid:82) . Remark . If p ˆ f , ˆ g q is a good pair, then we can consider the quotient M ˆ f “ Ă M { ă ˆ f ą which is a solid torus trivially foliated by the leaves of the induced folia-tion F ˆ f . The leaf space L F ˆ f “ M ˆ f { F ˆ f is a circle where η ˆ f , the action induced byˆ f in the quotient, acts as a homeomorphism. The same can be done to produce M ˆ g “ Ă M { ă ˆ g ą .We will mostly have in mind the following two examples on which our resultswill be applied and eventually specialize to these cases: Example . Let M be a closed 3-manifold and F be a minimal foliation in M preserved by a diffeomorphism f : M Ñ M homotopic to the identity. It followsfrom [BFFP , Corollary 4.7] that if we consider r f to be a good lift of f (i.e. thelift obtained by lifting a homotopy to the identity, cf. [BFFP , Definition 2.3])then either every leaf of r F is fixed by r f or F is (cid:82) -covered and uniform and r f acts as a translation on the leaf space L of r F . Since r f is a bounded distancefrom the identity, one can easily show that r f acts as the identity on S univ , when F is (cid:82) -covered. Moreover, as r f commutes with all deck transformations (whichare lifts of the identity that clearly preserves F ), it is enough to find a decktransformation γ which does not fix any leaf of r F to obtain a good pair p r f , γ q .Note that such deck transformations are quite abundant (see for instance [BFFP , §
8] and [BFFP , §
10] for the case where M is a hyperbolic 3-manifold). In thispaper we will later consider this setting when r F acting as a translation and M isa hyperbolic 3-manifold to prove Theorem B. Example . Let f : M Ñ M be a diffeomorphism of a Seifert manifold M withhyperbolic base and preserving a horizontal (branching) foliation F . Horizontalmeans that the Seifert foliation is isotopic to one which is transverse to F . Inparticular F is (cid:82) -covered and uniform. Since M has hyperbolic base, it followsthat, up to taking the square, f preserves the center of π p M q which correspondsto the circle fibers and is generated by a deck transformation γ . Note that γ acts as the identity on S univ . Moreover, if r F is the lift of F to Ă M it follows that γ does not fix any leaf of r F because F is horizontal. Moreover, any lift r f of f commutes with γ . If one fixes a lift r f , it follows that for large enough m ą p γ m r f , γ q is a good pair. This setting will be considered when the action inthe base is pseudo-Anosov to prove Theorem C. Notation . Given a good pair p ˆ f , ˆ g q and m, n integers we denote the by P tothe diffeomorphism P “ ˆ g m ˝ ˆ f n of Ă M and by P the induced homeomorphismof S univ (cf. Proposition 2.5). The values of m, n will be clear in the context. Formally, we need to take the approximating foliation to define the leaf space, but one canalso define L F ˆ f by using the action of ˆ f in L F . Super attracting points.
In this subsection we study what happens whena good pair has a super attracting fixed point in the universal circle (cf. Defini-tion 2.6) and how this forces some behavior in Ă M . We remark that such usefulinformation in Ă M from the action at infinity cannot in general be obtained froma merely attracting fixed point in S univ .We first need to describe some natural ‘neighborhoods’ of points of S univ inside Ă M adapted to a good pair p ˆ f , ˆ g q .We will assume in all this subsection that p ˆ f , ˆ g q is a good pair preserving F and that ˆ g is the element of the pair which acts as the identity on S univ .Recall that for any L in r F we denote Θ L : S univ ÞÑ S p L q the map thatassociates to a point in S univ its representative in S p L q .We will need to introduce some notations. Given an interval I of S univ con-taining ξ in its interior and L P r F , we denote by (cid:96) LI the geodesic in L joining theendpoints of Θ L p I q .Given a leaf L P r F we denote by L I to the closure of the connected componentof L z (cid:96) LI whose closure in L Y S p L q contains Θ p I q . Given b ą L ` bI Ă L (resp. L ´ bI ) to the union of L I with the b -neighborhood of (cid:96) LI (resp. thepoints in L I at distance larger than b from (cid:96) LI ). Definition 3.6.
Given ξ P S univ we say that an open set U of Ă M is a neighborhood of ξ if it is ˆ g invariant and for every L P r F we have that U X L contains L I p L q for some I p L q Ă S univ open interval containing ξ , and I p L q varying continuouslywith L . In addition we say that an unbounded sequence x n P Ă M converges to ξ P S univ if for every U neighborhood of ξ there is n such that if n ą n then x n P U .There is a lot of freedom in the definition of these neighborhoods of points ξ in S univ . Notice in particular that we require that the neighborhood is ˆ g invariant(where g is the identity). This requirement is necessary for some technicalresults later on to hold. Proposition 3.7.
Let p ˆ f , ˆ g q be a good pair and P “ ˆ g m ˝ ˆ f n so that ξ in S univ is a super attracting fixed point of P . There is a neighborhood U of ξ in Ă M sothat P p U q Ă U and for any x in U then P i p x q converges to ξ when i Ñ 8 . We will first construct a family of neighborhoods of ξ depending on openintervals I so that ξ P I Ă S univ , and a given leaf L P r F . Lemma 3.8.
Fix a leaf L P r F , then, for every open interval I Ă S univ there is I open arbitrarily close to I in S univ satisfying the following: we can define anopen set U I which is a neighborhood of any σ P I (cf. Definition 3.6) and suchthat L X U I “ L I . Moreover, there exists b ą such that for every E P r L, ˆ g p L qs we have that E ´ b I Ă U I X E Ă E ` b I Proof.
Since the result is in the universal cover, we can assume by taking adouble cover that F is transversely orientable. Then as we mentioned before,Burago-Ivanov proved that F is approximated by an actual foliation F ε . Theuniversal circles of F and F ε are canonically homeomorphic, under and equivarianthomeomorphism. Given a leaf E of r F there is an associated leaf E of r F ε .The foliation F ε has leaves with curvature arbitrarily close to ´
1. A contractingdirection in a leaf of E of r F ε is an ideal point y of E so that a geodesic ray r from a basepoint x in E to the ideal point y satisfies that all nearby leaves in ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 17 one side of E contracts towards E along r . Thurston proved (see [Fen , § E of r F ε the set of contracting points from E is dense in S p L q . In the first case for every δ ą E so that neary leavesstay always less than δ from E in these directions. The contracting points orpoints which are δ close to nearby leaves project down to similar points of thebranching foliation F .There is k ą g of any geodesic (cid:96) in a leaf E of r F is a k quasigeodesic in ˆ g p E q . Let b be a global constant so that if (cid:96) is a k quasigeodesic in a leaf E of r F , then (cid:96) is at most b { E with same ideal points as (cid:96) . Fix δ ą F . Now let δ ą , δ ăă δ so that if two leaves E , E of r F are within δ of each other along a geodesic β of E , then E , E arewithin δ of each other in a neighborhood of size b of β . This is why we use theapproximating foliation, rather than the branching foliation.So fix the leaf L as in the statement of the lemma. Given ˆ g p L q we find idealpoints z , z in S p ˆ g p L qq arbitrarily close to the endpoints Θ ˆ g p L q pB I q so that raysin ˆ g p L q in the direction of z i are δ close to all nearby leaves E of r F in r L, ˆ g p L qs .Let y i “ p Θ ˆ g p L q q ´ p z i q .Consider (cid:96) the geodesic in L with ideal points Θ L p y i q . Consider in ˆ g p L q thegeodesic β with same ideal points as ˆ g p (cid:96) q . Then β, ˆ g p (cid:96) q are at most b { g p L q . Choose E in r L, ˆ g p L qs which is at most δ from ˆ g p L q along β . One can do this for some rays of β in either direction by the choice of I .Then by choosing E closer to ˆ g p L q if necessary, one can choose this for the wholegeodesic β . Let B be the b { β in ˆ g p L q . Then B is δ near E for any E in r E, ˆ g p L qq .In E let (cid:96) E be the geodesic with ideal points Θ E p y i q . The foliation r F is aproduct in the δ neighborhood of B , hence one can continuously chooose curves (cid:96) G for G in r E, ˆ g p L qs so that:(i) (cid:96) G is a quasigeodesic in G ,(ii) (cid:96) G has ideal points Θ G p y i q ,(iii) (cid:96) G is within b of the geodesic β G in G with ideal points Θ G p y i q .(iv) (cid:96) E is the geodesic with ideal points Θ E p y i q .(v) (cid:96) ˆ g p L q “ ˆ g p (cid:96) L q .Now for G in r L, E s let (cid:96) G be the geodesic with ideal points Θ G p y i q .This defines the neighborhood U I for G in r L, ˆ g p L qs . Then iterate by ˆ g toconstruct all of U I . By construction U I satisfies the last property of the lemma.In addition L X U I “ L I .Finally we check the first property of the lemma. For any G leaf of r F , there isa unique n so that E “ ˆ g ´ n p G q is in r L, ˆ g p L qq . Then U I X G is ˆ g n p E X U I q . Theset E X U I is bounded by a uniform quasigeodesic in E , with endpoints Θ E pB I q .Hence ˆ g n p E X U I q “ G X U I are also bounded by uniform quasigeodesics withideal points Θ G pB I q , because ˆ g n is a quasi-isometry between leaves. Hence forany σ in I , there is J open subinterval of I containing σ so that the set U I X G contains G J for all G in r L, ˆ g p L qs .This finishes the proof of Lemma 3.8. (cid:3) Remark . Note that if E R r L, ˆ g p L qs one cannot ensure the containment andinclusion with the same constant b : this is because one applies iterates of thequasi-isometry ˆ g , whose quasi-isometry constants get worse with iteration.This means that given σ P I it is not a priori true that there is a fixed interval J with σ P ˚ J and J Ă I with G J Ă U I X G for all G in r F . Proof of Proposition 3.7.
Given I Ă S univ , U I as constructed in the previouslemma, and E P r F , we denote A IE “ E X U I .We claim that if I as above is a sufficiently small interval around ξ , then thereare smaller intervals J around ξ such thatˆ g k ˝ P p A IE q Ă A JE for all E in r L, ˆ g p L qs . Here E P r L, ˆ g p L qq is the image of E by ˆ g k ˝ P and k P (cid:90) is defined uniquely so that ˆ g k ˝ P p E q P r L, ˆ g p L qq . This will complete the proofbecause: the formula above shows that P p A IE q Ă U J for all E in r L, ˆ g p L qs . Thefact that P p U q Ă U then follows from the facts below:1) r L, ˆ g p L qs is a fundamental for the action of ˆ g on Ă M ,2) P commutes with ˆ g ,3) Both U I and U J are ˆ g invariant.To get the property above, first note that the value of k is uniformly boundedin r L, ˆ g p L qs and so one gets that the quasi-isometric constants of the map ˆ g k ˝ P : E Ñ E are uniformly bounded independently on E P r L, ˆ g p L qs (where the k depends on the particular leaf E ). It follows that there exists b ą J Ă S univ if we denote by Z “ P p J q we have thatˆ g k ˝ P p E ` b J q Ă p E q ` b Z For every E in r L, ˆ g p L qs .Now we use the property of ξ being super-attracting for the map P . Since ˆ g acts as the identity on S univ , then ξ is super attracting for p ˆ g k ˝ P q .For each fixed k one has that p ˆ g k ˝ P q “ P . Therefore one gets that forsmall enough intervals I around ξ the image Z “ P p I q verifies that the distancebetween the geodesics (cid:96) EI and (cid:96) EZ are much larger than 2 b ` b for every E Pr L, ˆ g p L qs . Here again we use that r L, ˆ g p L qs is a compact interval in the leaf spaceof r F . Hence we can choose J interval in S univ around ξ and U J as defined in theprevious lemma, so that (cid:96) LJ separates (cid:96) LI from (cid:96) LZ , and p E q ` b Z Ă p E q ` J Ă U I X E for all E in r L, ˆ g p L qs . Let U “ U I . This proves that P p U q Ă U .In addition one can choose the starting I small enough, so that in the proofabove the distance from any point in (cid:96) IE to any point in (cid:96) JE is bigger than aconstant b ąą b ` b for all E in r L, ˆ g p L qs .This holds for smaller I as well. In particular it holds for J . Hence the distancein E (for any E in r L, ˆ g p L qs ) from any point in (cid:96) IE to any point in P p U I q X E is at least 2 b , and similarly for any positive P i iterate it is at least ib . Thisimplies that for any neighborhood V of ξ there is i ą P i p U I X r L, ˆ g p L qsq Ă V. The ˆ g invariance of the sets U I and V then implies that P i p U I q Ă V . Hence forany x in U then P i p x q converges to ξ when i ÞÑ 8 . This completes the proof ofProposition 3.7. (cid:3)
ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 19
Definition 3.10.
Given a good pair p ˆ f , ˆ g q so that ξ P S univ is super attracting(resp. super repelling) for P “ ˆ g m ˝ ˆ f n . We define the basin of attraction (resp. basin of repulsion ) of ξ to be the set of points x in Ă M such that P k p x q Ñ ξ as k Ñ `8 (resp. k Ñ ´8 ) understood as in Definition 3.6.Proposition 3.7 says that a super attracting point (which is defined only bythe action on S univ ) has a non-trivial basin of attracting which is a neighborhoodof the super attracting point. Remark . Let p ˆ f , ˆ g q be a good pair and assume that ˆ g act as the identity on S univ . It follows that if a point ξ P S univ is super-attracting for a lift P “ ˆ g m ˝ ˆ f n ,then n ‰ ξ will be super-attracting for every lift ˆ g k ˝ ˆ f (cid:96)n if (cid:96) ą (cid:96) ă Pseudo-Anosov pairs.
We can now define a technical object that will becentral in our proofs:
Definition 3.12.
A good pair p ˆ f , ˆ g q is a pseudo-Anosov pair (or pA-pair ) ifthere is n, m P (cid:90) such that if P “ ˆ g m ˝ ˆ f n then the homeomorphism P in theuniversal circle S univ (cf. Notation 3.5) has exactly 2 p fixed points, all of whichare alternatingly super-attracting and super-repelling. Here p is an integer ě p “ regular pA-pair and if p ě prong pA-pair . We denote by I p P q “ ´ p the index of the pseudo-Anosov pair. W J ej T E n.n se P P
Figure 1.
A pseudo-Anosov pair with p “ We next state a result which extends [BFFP , Proposition 8.1] and [BFFP , § Proposition 3.13.
Let p ˆ f , ˆ g q be a pseudo-Anosov pair and P “ ˆ g m ˝ ˆ f n be a liftwith m, n ‰ , satisfying the conditions of Definition 3.12.Then there exists a closed set T P Ă Ă M which is invariant under ˆ f and ˆ g . Theset T P intersects every leaf L of r F in a compact set T P X L which consists of theset of points which are not in the basin of attraction of any attracting point of P or the basin of repulsion of any repelling point of P .Moreover, if there is a leaf L P r F such that P p L q “ L , then the total Lefschetzindex of the compact invariant set T P X L is I p P q the index of P . The set T P is called the core of the pair. It is the complement in Ă M of whatone detects by looking at the action at infinity.We first define the basins of attraction. Let p ˆ f , ˆ g q be a pseudo-Anosov pair withˆ g acting as the identity on S univ , and let P “ ˆ g m ˝ ˆ f k as in Definition 3.12. Let t a , . . . , a p u and t r , . . . , r p u be the super attracting and super repelling points of P on S univ . We define T ` P (resp. T ´ P ) as the set of points which is not in thebasin of attraction of any of the points a , . . . a p P S univ (resp. not in the basisof repulsion of any of the points r , . . . , r p P S univ ). Let T P “ T ` P X T ´ P . Proposition 3.13 follows from applying the following consequence of Proposi-tion 3.7 that we state precisely for future use and prove below.
Proposition 3.14.
Let p ˆ f , ˆ g q be a pseudo-Anosov pair with ˆ g acting as the iden-tity on S univ , and let P “ ˆ g m ˝ ˆ f k as in Definition 3.12. Then: (i) The set L z T ` P (resp. L z T ´ P ) is non empty and open. (ii) For each L in r F , L X T P ‰ H . (iii) For any L in r F , then L X T P is compact. In addition T P { ă ˆ g ą , T P { ă ˆ f ą are compact. (iv) For every ξ P S univ zt r , . . . r p u (resp. ξ P S univ zt a , . . . , a p u ) if we denote a i (resp. a i ) to be the point such that P n p ξ q Ñ a i (resp. P ´ n p ξ q Ñ r i )then, there exists a neighborhood U ξ of ξ in Ă M contained in the basin ofattraction of a i (resp. basin of repulsion of r i ).Proof. Item (i) follows directly from Proposition 3.7 because it proves that thebasins of attraction and repulsion of each point in t a , ..., a p , r , ..., r p u are open,non empty sets in Ă M .Next we prove item (iv). Fix L in r F . Let ξ not one of the r i . Then ξ is inthe basin of attraction of some attracting point under P , assume without lossof generality it is a . Let U “ U I be a neighborhood of a constructed as inProposition Lemma 3.7. Let i ą P n p ξ q is in the interior of I . Thenas in the proof of Proposition 3.7, there is J open interval containing P i p ξ q sothat E b ` b J Ă U for all E in r L, ˆ g p L qs . We can construct a set U J in r L, ˆ g p L qs as in Proposition 3.7 so that U J X E Ă U for all E in r L, ˆ g p L qs . Then iterate byˆ g to produce U J . It is a neighborhood of P i p ξ q which is contained in the basisof attraction of a . Then P ´ i p U J q is the desired neighborhood of ξ contained inthe basis of attraction of a . This proves (iv).To obtain item (iii) we do the following. Let ξ in S univ not one of t r i u . There is a an attracting point of P so that ξ is in the basis of attraction of a under P .Fix a neighborhood U I of a contained in the basis of attraction of a under P as provided in item (iv). There is i so that P i p ξ q P I . Fix L in r F , let τ “ Θ L p ξ q .The above shows that P i p τ q is an interior point of Θ P i p L q p I q . In particular thereis a neighborhood V of τ in L Y S p L q so that P i p V X L q Ă p U I X P i p L qq . Thisis because of the definition of the neighborhoods U I . Similarly there is j so that P j p ξ q is in J where U J is contained in a repelling neighborhood of a repellingpoint r of P . Both of these facts together imply that L X T P is compact for any L in r F . In the argument above one can take the neigh-borhood in Ť E P Z p E Y S p E qq , where Z is any compact interval in the leaf spacewith L in the interior. This shows that T P { ă ˆ g ą , T P { ă ˆ f ą are compact.Finally we prove item (ii). Fix L in r F . ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 21
Fix a union V of neighborhoods of the points r , ..., r p , so that P ´ p V q Ă V .For any n ą
0, the set A n “ p T ` P ´ V q X P ´ n p L q is non empty. Otherwise the basins of attraction of different a i , a j intersect, whichis impossible. Choose x n in A n and let y n “ P n p x n q which is in L . In addition y n is not in V and y n is in T ` P . Therefore y n is in a compact set of L . Take asubsequence y n i converging to y in L . If y is not in T ´ P then there is n ą P ´ n p y q is in V , so a neighborhood W of y so that P ´ n p W q Ă V for any n ą n . Assume all y n i are in W . But this contradicts that P ´ n i p y n i q “ x n i arenever in V .This contradiction shows that y is in T ´ P . Since x n is in T ` P then y n is also in T ` P and y is in T ` P . It follows that y is in T P so T P X L “ H . This finishes theproof of (ii).This finishes the proof of the proposition. (cid:3) W J ej T E n.n se T ´ P T ` P T P Figure 2.
The core of the pA pair.
Since for pA pairs (and appropriate choices of integers n, m with P “ ˆ g m ˝ ˆ f n )all fixed of P in S univ are super attracting and super repelling, we will disposethe use of the word super when it is clear that we are considering a pA pair andcall the points attracting and repelling. Addendum 3.15.
In the setting of Proposition 3.14 we have the following: forevery family of attracting neighborhoods U a i of the attracting points a i and L in r F , there is some R ą such that outside a ball of radius R in E we havethat T ´ P X E is contained in those neighborhoods, for any E in r L, ˆ g p L qs . Thesymmetric statement holds for the repelling points and T ` P . In particular, T P X L is contained in a ball of radius R inside L . This is obvious because the quotient T P { ă ˆ g ą is compact.An argument very similar to the proof of item (ii) of the previous propositionyields the following result which will be useful in the future. The map ˆ g acts freelyand properly discontinuously in Ă M , hence Ă M { ă ˆ g ą is a manifold N (cf. Remark r F induces a foliation F N in N , whose leaves are homeomorphicto planes and the leaf space of F N is the circle. Let π N : Ă M Ñ N be the projectionmap. We say that a sequence x n in Ă M converges to T P if π N p x n q converges to T P { ă ˆ g ą , in the sense that for any neighborhood Z of T P { ă ˆ g ą in N then π N p x n q is eventually in Z . Lemma 3.16.
Under the hypothesis of Proposition 3.7 let y in T ` P . Then P n p y q converges to T P as n Ñ 8 .Proof.
We use the setup in the proof of item (ii) of the previous proposition. Inparticular let V be a union of neighborhoods of the repelling points of P so that P ´ p V q Ă V .Let P N be the induced map by P in N . Let z “ π N p y q . Assume that P n p y q does not converge to T P . Then there is a neighborhood Z of T P { ˆ g and n i Ñ 8 ,with P n i N p z q always not in Z . There is n ą n ą n then P n p y q is notin V , hence P nN p z q is not in V { ă ˆ g ą . By the addendum, P nN p z q is in a compact setin N for n ą n . Hence up to another subsequence we may assume that P n i N p z q converges to z . Notice that z is not in T P { ă ˆ g ą .Lift z to x in Ă M . Then x is not in T P hence as in the proof of item (ii)of the previous proposition there is a neighborhood W of x and j integer sothat if j ď j then P j p w q is in V for any w in W . For any i big P n i N p z q is in W { ă ˆ g ą , hence P n i ` j N p z q is in V { ă ˆ g ą . This contradicts the fact that P nN p z q is notin V { ă ˆ g ą for n ą n . This contradiction finishes the proof. (cid:3) Abundance of pseudo-Anosov pairs.
In this section we specialize to thecases described in examples 3.3 and 3.4: That is, we say that p f, F q verifies the commuting property if f : M Ñ M is a diffeomorphism preserving an (cid:82) -covereduniform foliation F by hyperbolic leaves and if one of the following conditionsholds:(i) There is a lift ˆ f to Ă M which commutes with all deck transformations and does not fix any leaf of r F .(ii) There is a deck transformation γ which commutes with all deck transfor-mations and does not fix any leaf of r F .The assumption that p f, F q has the commuting property implies on the onehand that it admits good pairs of the form p ˆ f , γ q with ˆ f a lift of f and γ P π p M q a deck transformation (i.e. a lift of id : M Ñ M ) and on the other that one canconstruct new good pairs out of others. Definition 3.17.
A good pair p ˆ f , γ q for a p f, F q with the commuting propertywill be said to be admissible if either ˆ f commutes with all deck transformations,or γ is in the center of π p M q .In the first case, the good pair property is verified since ˆ f acts as the identityon S univ , and in the latter, it is γ that acts as the identity on S univ . In both casesthis happens because if a map commutes with all deck transformations then thismap is a bounded distance from the identity in Ă M (see also [BFFP ]). In particular, if f is homotopic to the identity. In particular, M is Seifert with hyperbolic base (because the leaves of F are hyperbolic)and γ corresponds to the center of π p M q generated by the element corresponding to the fibers.This in particular implies that F is horizontal, and so S univ identifies with the boundary of theuniversal cover of the base. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 23
Definition 3.18.
Let p f, F q have the commuting property and let p ˆ f , γ q be anadmissible good pair. We say that p ˆ f , γ q is conjugate to p ˆ f , γ q by η P π p M q ifwe have that p ˆ f , γ q “ p η ´ ˝ ˆ f ˝ η, η ´ ˝ γ ˝ η q . Note that if p ˆ f , γ q is a pA pair (resp. regular pA pair) then every good pairconjugate to p ˆ f , γ q also is a pA pair (resp. regular pA pair).The following result shows that there are plenty of pA-pairs with good prop-erties. In our specific settings, we could obtain this directly, but here we give aunified proof. Proposition 3.19.
Let p f, F q with the commuting property, p ˆ f , γ q be an ad-missible pA-pair and let J Ă S univ be an open interval. Then, there exists anadmissible pA-pair p ˆ f , γ q conjugate to p ˆ f , γ q such that it has all its fixed pointsin the interior of J .Proof. We will apply [FP , Lemma 5.4] stating that for every pair of disjoint opensets U and V in S univ there is a deck transformation η such that the action of η on S univ maps the complement of U in the interior of V . Denote by P “ γ m ˝ ˆ f n with n, m not both equal to 0 and denote P the action on S univ .Now, pick a open interval U disjoint from all fixed points of P and a decktransformation η which maps the complement of U inside J .Now, if one considers the map η ˝ P ˝ η ´ it follows that it has all its fixedpoints inside J . Since the map is conjugated by a deck transformation, it followsthat the points are super attracting/repelling. This is because deck translationsacts in a H¨older way on S univ .Since p f, F q has the commuting property, then either p ˆ f , η ˝ γ ˝ η ´ ) or p η ˝ ˆ f ˝ η ´ , γ q make an admissible pair for p f, F q . (cid:3) Pseudo-Anosov pairs and sub-foliations
We will assume that there is a one-dimensional branching foliation T subfo-liating F which is f -invariant (recall the definition at the end of § r T the lift of T to the universal cover. Recall from the previous section thatwhenever p ˆ f , γ q is a pseudo-Anosov pair we will take P to be some P “ γ m ˝ ˆ f n which has a finite number of fixed points alternatingly super attracting and superrepelling in S univ . If we do not choose explicitely the values of n, m it will meanthat any choice with this property will work.4.1. Landing points.
Given a leaf c P r T we say that ˆ c is a ray of c if it is aconnected component of c zt x u for some x P c . Since the leaves of r T are properlyembedded in leaves of r F then every ray of c Ă L P r F accumulates only in someconnected subset of S p L q .Using the dynamics of pseudo-Anosov pairs one deduces the following simpleproposition that we will use several times in the paper. In this result we we usethe foliation F N in N “ Ă M { ă ˆ g ą . Proposition 4.1.
Let p ˆ f , ˆ g q be a pseudo-Anosov pair and let ˆ c be a ray in aleaf of r T which accumulates in an interval J Ă S p L q . Then, J contains the Θ L images of at most two fixed points of P in S univ . Note that by the commuting property we have that either η ´ ˝ ˆ f ˝ η “ ˆ f or η ´ ˝ γ ˝ η “ γ . Proof.
We assume that ˆ g acts as the identity on S univ . If the interval J intersectsthree such points we can assume without loss of generality that two of them (wecall them a , a , points in S univ ) are attracting while one (called r ) is repellingand between the two attracting ones there are no other fixed point of P .Fix neighborhoods U a and U a of a and a in the respective basins of at-traction given by Proposition 3.7. For those neighborhoods there is a sequence (cid:96) , . . . , (cid:96) k , . . . of arcs of ˆ c joining the neighborhoods U a and U a . Fix a repellingneighborhood V r of the form U I of r so that I has endpoints y , y . We assumethat y i is in the interval p a i , r q of S univ . Then P i p y i q converges to a i as i Ñ 8 .Let b i “ Θ L p y i q . For i big (cid:96) i has a subsegment e i in V r X L connecting a pointvery near b in L Y S p L q to a point very near b in L Y S p L q . These points arein the basins of attraction of a , a respectively. We claim that e i intersects T ` P .If not then e i is contained in the union of basis of attraction of attracting points,but the endpoints are contained in distinct basis of attraction. Since the basis ofattraction are open sets this contradicts the connectedness of e i .We consider the manifold N “ Ă M { ă ˆ g ą as in Lemma 3.16 (see also Remark3.2). Let T N be the foliation induced by r T in N . As in Lemma 3.16 considera fixed neighborhood Z of T P { ă ˆ g ą in N , but now with compact closure. Coverthe closure of Z by finitely many foliated boxes of F N and T N all with compactsupport. Since the leaves of F N are planes, and F N is a fibration over the circlethe following happens: we can choose the foliated boxes small enough so that aleaf of T N can only intersect each of these foliated boxes in a single component.Since (cid:96) i intersects T ` P , Lemma 3.16 implies that there is some k i such that if k ą k i the map P k will then map the arc (cid:96) i to a curve intersecting π ´ N p Z q .We can apply this several times to all the arcs, we find a sufficiently largenumber of subarcs of a large iterate of ˆ c that when projected to N they allintersect Z . If there are sufficiently many, then more than two have to intersecteda product box of T N in Z .This contradicts the fact that each curve of r T N can only intersect a localproduct box as above in a unique connected component.This produces a contradiction and proves the proposition. (cid:3) M Figure 3.
Proof of landing. The iterates P n push the leaves awayfrom the middle point and into a compact part (when projectedto N ). Throughout the remainder of this section we will consider p f, F q withthe commuting property (cf. § ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 25
We will assume that there is a one-dimensional branching foliation T subfoliat-ing F which is f -invariant (recall the definition at the end of § c of a curve c P r T lands in a point if there exists ˆ c P S p L q so that theclosure of ˆ c in L Y S p L q is ˆ c Y ˆ c . In other words the ray ˆ c accumulates only onˆ c .Using the previous proposition, we deduce the following: Proposition 4.2. If f : M Ñ M preserves an (cid:82) -covered and uniform foliationby hyperbolic leaves F which is subfoliated by a one-dimensional foliation T andadmits a pseudo-Anosov pair p ˆ f , γ q then every ray of r T lands in a point.Proof. It is enough to show that given a pseudo-Anosov pair p ˆ f , γ q and an interval J Ă S univ there exists another pseudo-Anosov pair so that J contains three ormore fixed points of it, in order to apply Proposition 4.1. But the fact we needfollows from Proposition 3.19. (cid:3) Notation . In Ă M we can orient leaves of r T , and we will fix an orientation.Given a leaf (cid:96) P r T in a leaf L P r F both of whose rays land in a point we denoteby (cid:96) ` and (cid:96) ´ in S p L q the landing points of the positive and negative ray (withrespect to the orientation and a given point x P (cid:96) which is not relevant for thedefinition of (cid:96) ˘ ).4.2. Pseudo-Anosov pairs with periodic leaves.
We let p ˆ f , γ q be a pA-pairand we will assume that:(i) There is a leaf L P r F which is fixed by P “ γ m ˝ ˆ f n for some m ‰ n ą P in S univ has 2 p fixed points which are alternativelysuper attracting and super repelling (with p ě c P r T X L be a leaf which is fixed by P and x P c . Write c “ c Y t x u Y c where c and c are the two connected rays of c defined by x . Suppose that c has ideal point c ` in S p L q and c has ideal point c ´ . We say that c is coarselyexpanding (resp. coarsely contracting ) if there is a compact interval I of c suchthat for every compact interval J of c Y t x u there is k ą P ´ k p J q Ă I (resp. P k p J q Ă I ). These rays already played a prominent role in the argumentsof [BFFP ]. The next result should be compared with the results in [BFFP , § Proposition 4.4.
Given a center curve c P r T X L which is fixed by P , then Θ ´ L p c ` q , Θ ´ L p c ´ q are fixed by P in S univ . Moreover, if Θ ´ L p c ` q is an attracting(resp. repelling) point of P in S univ then the ray c is coarsely expanding (resp.contracting).Proof. This is direct from Proposition 3.7. See also [BFFP , § (cid:3) We now give a definition that we will use several times since we will be ableto establish this strong property in the partially hyperbolic setting:
Definition 4.5.
A pair p f, F q has the periodic commuting property if it has thecommuting property (cf. § p ˆ f , γ q for p f, F q there exists k ą m P (cid:90) zt u and a leaf L P r F which is fixed by P “ γ m ˝ ˆ f k . Notation . Whenever p f, F q has the periodic commuting property and p ˆ f , γ q is an admissible pA-pair, the lift P will denote a lift P “ γ m ˝ ˆ f k with m P (cid:90) zt u and k ą P fixes some leaf L P r F and such that P acting on S univ hasfixed points, all of which are either super attracting or super repelling. Proposition 4.7. If p f, F q has the periodic commuting property and p ˆ f , γ q isany admissible pA-pair, then for every L P r F one has that P m p L q converges as m Ñ ˘8 to a leaf which is fixed by P .Proof. The hypothesis implies that P fixes a leaf E of r F . But then it also fixes γ i p E q for any i P (cid:90) . For any L leaf of r F it is contained in r γ i E, γ i ` E s for some i in Z which implies the result. (cid:3) The following property will be used several times: M Figure 4.
A configuration.
Proposition 4.8.
Let p f, F q have the periodic commuting property preservinga one dimensional branching foliation T that subfoliates F . Let p ˆ f , γ q be anadmissible regular pA-pair with attracting points a , a for a lift P “ γ m ˝ ˆ f n (cf.Notation 4.6). Assume that there is a leaf (cid:96) of r T X L which has a segment I Ă (cid:96) with the property that both extreme points of I belong to the basin of attractionof a and so that I intersects the basin of attraction of a . Then, there is aleaf E P r F fixed by P which has at least two disjoint leaves (cid:96) and (cid:96) from r T joining Θ E p a q and Θ E p a q and (cid:96) , (cid:96) fixed by P . In particular, (cid:96) and (cid:96) areboth coarsely expanding for P .Proof. Let E be the limit of P k p L q as k Ñ 8 given by Proposition 4.7 which isfixed by P .Call x and x the endpoints of I so that I is oriented from x to x . Take y P I which belongs to the basin of attraction of a . It follows that I “ I Y I where I is the segment oriented from x to y and I the segment oriented from y to x .Let r , r be the other fixed points of P , which are both repelling, and a , r , a , r circularly ordered in S univ . Let V , V be neighborhoods of type U I of r , r respectively so that P ´ p ¯ V i q Ă V i , given by Proposition 3.7. Let I i be the interval of S univ defined by V i and containing r i .For k ą P k p I i q cannot intersect V or V , therefore the sequence p P k p I qq cannot escape compact sets in Ă M as k Ñ 8 . This is because P k p L q converges to E and P k p I i q intersects the basis of attraction of both a and a . ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 27
Hence the sequence p P k p I qq converges to some family of leaves in r T in E . Theleaves in the limit have must land. The set of landing points of all such limitleaves is invariant under P (since E is invariant under P ). In addition the setof landing points of these limit leaves cannot intersect Θ E p I q or Θ E p I q because I , I are expanding intervals under the action of P . Therefore the only possiblelimit points of the landing leaves must be Θ E p a q and Θ E p a q . See Figure 4 fora depiction of this situation.Since P k p I q has endpoints in neighborhoods of a and a there must be a limitleaf which has a and a as landing points (as opposed to both landing pointsbeing a or a ). In addition this leaf is oriented going from a to a . Similarly inthe limit of p P k p I qq there must be at least one leaf of r T oriented from a to a .The family of such limit leaves is closed, ordered, and avoids neighborhoods ofthe repellers of P ; so we can consider the two outmost of them and these mustbe fixed by P . Moreover, since they are oriented in a different direction, theseleaves cannot be close and therefore are disjoint. (cid:3) Indeed we get a further property:
Addendum 4.9.
In the setting of Proposition 4.8 we further obtain that in E there is at least one leaf (cid:96) of r T in E between (cid:96) and (cid:96) so that both endpointscoincide with either a or a .Proof. Consider the leaves (cid:96) , (cid:96) in E obtained in Proposition 4.8. These leavesform the boundary of an infinite band B in E which accumulates only in a and a . Therefore any leaf of r T contained in B can only accumulate in these twopoints. Assume by way of contradiction that no such leaf has both endpoints a or both a . Then every leaf has one ideal point in a and one in a . As aconsequence the set of leaves of r T X E between (cid:96) and (cid:96) has an order makingit order isomorphic to an interval. Moreover, each such leaf has an orientationeither from a to a or from a to a . Since the orientations of (cid:96) and (cid:96) differ,this is a contradiction. Therefore it cannot happen that every leaf in between (cid:96) and (cid:96) has different endpoints. (cid:3) Shadows and visual measure.
Let p f, F q with the periodic commutingproperty (Definition 4.5) preserving a one dimensional branching foliation T thatsubfoliates F .In some cases it is possible to control the visual measure of a center arc froma point in a leaf L P r F if one can exclude certain configurations. For a point x P L P r F and a subset X Ă L we call the shadow of X from x to the set ofpoints in S p L q – T x L corresponding to geodesic rays from x intersecting X . Definition 4.10.
We say that T has small visual measure in F if for every ε ą R ą x P L P r F and I is a segment of r T X L at distancelarger than R from x , then the shadow of I from x has visual measure smallerthan ε in S p L q – T x L (cf. § d L p x, y q ą R for any y in I . Proposition 4.11.
Assume that there is a regular admissible pA-pair for p f, F q and that T does not have small visual measure in F . Then, there is a regularpA-pair p ˆ f , γ q and a leaf L P r F fixed by P (cf. Notation 4.6) which has at leasttwo disjoint leaves of r T each fixed by P and whose landing points are Θ L imagesof distinct attracting fixed points of P in S univ . In particular, these leaves of r T are both coarsely expanding for P . M Figure 5.
The shadow.
Proof.
By assumption there is (cid:15) ą
0, and there are points x n in leaves L n P r F such that there are segments I n of leaves of r T X L n at distance bigger than n from x n and whose shadow in S p L n q has visual measure larger than ε . Decktransformations act as isometries on leaves of r F . Hence up to applying decktransformations and a subsequence we can assume that x n converges to a point x . We can assume that L n converges to L (notice F is a branching foliation soa priori all L n could contain x ).Let J n be the shadow of I n on S p L n q . Up to another subsequence we canalso assume that the intervals J n in S p L n q converge to an interval J of visualmeasure larger than ε in S p L q . We can assume without loss of generality that J ‰ S p L q by taking shorter segments I n .Using Proposition 3.19 we can consider p ˆ f , γ q an admissible regular pA-pairsuch that it has all of its fixed points in the interior of J “ Θ ´ p J q . Call theattracting points a , a and the repelling ones r , r . Since J ‰ S univ we canorder these points in S univ up to renumbering so that a is inside the segment J Ă J whose endpoints are r and r . Consider neighborhoods U a i and V r i ofeach in Ă M as in Proposition 3.7.For large enough n we have that the arcs I n contain subarcs S n Ă I n joining V r with V r and intersecting U a . We can also assume that S n are such that forsome points ξ , ξ in each connected component of J z J the segment S n intersectsthe neighborhoods U ξ i as in Proposition 3.14 ( iv ). In particular these points in U ξ i are in the basis of attraction of a (the other attracting point). Denote as S n and S n two segments of S n joining respectively U a with U ξ and U ξ .Now the result follows from Proposition 4.8. (cid:3) Remark . Note that the proposition admits a symmetric statement since itcan be applied to p ˆ f ´ , γ q which is a regular pA-pair for f ´ which also preserves F and T . So, under those assumptions there also exist a fixed leaf of r F withtwo distinct leaves of T being fixed and coarsely contracting. Disjointness of thecurves is important since we do not assume that T is a true foliation. This willallow us to rule out such behavior for centers in the partially hyperbolic setting.5. Pseudo-Anosov pairs and partially hyperbolic foliations
In this section f : M Ñ M will be a partially hyperbolic diffeomorphismpreserving two transverse branching foliations W cs and W cu . We denote by W s and W u the strong stable and strong unstable foliations respectively and by W c ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 29 the center (branching) foliation. We will assume that at least one of W cs or W cu is (cid:82) -covered and uniform and that some lift ˆ f acts as a translation on this leafspace. Many results will be stated for W cs but obviously work equally well for W cu .5.1. Periodic leaves for pseudo-Anosov pairs.
Here we restate a result from[BFFP , BFFP ] in the context of pseudo Anosov pairs. Proposition 5.1.
Assume that p ˆ f , γ q is a pA pair for the foliation W cs . Then,there exists n ą , m P (cid:90) ‰ and a leaf L P Ą W cs such that γ m ˝ ˆ f n p L q “ L .Proof. Under these conditions we proved in Proposition 3.14 (ii) that the set T P is non empty. The quotient of T P in Ă M { γ ´ that is, T P { γ is compact. Since γ is a deck transformation, the map ˆ f projects to a map in Ă M { γ which is partiallyhyperbolic and preserves the compact set T P { γ . After these facts are establishedthe proof is exactly the same as [BFFP , Proposition 10.3] (which itself uses[BFFP , Proposition 9.1]) or [BFFP , Proposition 4.1]. (cid:3) Remark . Notice that once one has this, one immediately deduces that f that W cs cannot be a true foliation (cf. [BFFP , Theorem B] and [BFFP , § W cs or W cu foliation cannot be dynamically coherent and will force that the map h in the definition of collapsed Anosov flow is not ahomeomorphism.As a consequence of Proposition 5.1 we deduce immediately that: Corollary 5.3. If p f, W cs q has the commuting property and has an admissiblepA-pair, then it has the periodic commuting property.Remark . Note that if p ˆ f , γ q is a pA pair and P “ γ m ˝ ˆ f n with n ą
0, themap P is a lift of a positive iterate of f therefore is partially hyperbolic andthe invariant bundles are exactly the lifts of those of f in M to Ă M (the stableswitches with the unstable if we take n ă Remark . Note that both W c and W s are one dimensional (branching) subfo-liations of W cs . By construction, it holds that W c is also a subfoliation of W cu (which is also a branching foliation) and therefore we know that in Ă M we havethat a curve of Ă W s cannot intersect the same leaf of Ă W c twice.5.2. Visual measure and distance of curves to geodesics.
Here we showthe following result which has validity beyond the context we are working in thispaper as it does not require a full set of pA pairs (defined later).
Theorem 5.6.
Let f : M Ñ M be a partially hyperbolic diffeomorphism pre-serving branching foliations W cs and W cu so that p f, W cs q has the commutingproperty (see subsection 3.4). Assume moreover that there is an admissible regu-lar pA pair for p f, W cs q . Then it follows that both W c and W s have small visualmeasure in W cs (cf. Definition 4.10). Moreover, there is R ą such that givena center leaf (cid:96) P Ă W c (resp. a stable leaf (cid:96) P Ă W s ) in L P Ą W cs if we denote by ˆ (cid:96) a segment a or ray of (cid:96) whose landing is either (cid:96) ´ or (cid:96) ` P L Y S p L q then thegeodesic segment or geodesic ray ˆ r of L joining either the endpoints of ˆ (cid:96) or thestarting point of ˆ (cid:96) with its landing point is contained in the R -neighborhood in L of ˆ (cid:96) . Remark . It is important to mention what this Theorem does not say. Inparticular, it does not ensure that the ray ˆ (cid:96) is contained in a neigborhood ofthe geodesic ray (in particular, it does not say that ˆ (cid:96) is a quasigeodesic). Later,we will use this result to show that under some more assumptions, all centercurves are quasigeodesics. This cannot hold for stable curves as there may besome stable curves which have both endpoints being the same (see eg. [BGHP]).However, the fact that the strong stables have small visual measure is somethingquite remarkable as they can be made to have tangent vectors arbitrarily closeto horocycles (see [BGHP]).Note first that the fact that curves from Ă W c and Ă W s land in leaves of Ą W cs isdirect from Proposition 4.2. To show that the visual measure of the arcs, raysor shadows is small we will use the following result about center curves that willalso be useful later: Lemma 5.8.
Let f be a partially hyperbolic diffeomorphism preserving branchingfoliations W cs and W cu so that p f, W cs q has the commuting property and there isan admissible regular pA pair p ˆ f , γ q and let P as in Notation 4.6 with P p L q “ L for some L P Ą W cs . Then, there cannot be two disjoint center curves c and c of Ă W c in L which are fixed by P and join the Θ L images of distinct attracting fixedpoints of P in S univ .Proof. Such center curves should be coarsely expanding by P by Proposition 4.4.This forces P to have at least one fixed point x in c . We look at s p x q the stablemanifold of x . It cannot intersect c since both s p x q and c are invariant by P and so is their intersection, which is a single point y . See Figure 6. Since c , c are disjoint y would be a fixed point of P in s p x q different from x ´ impossiblesince s p x q is a stable leaf and P is contracting in stables. Then, the ray of s p x q inthe connected component of L z c containing c must land in an attracting pointof P in S univ , this is impossible since s p x q is coarsely contracting (cf. Remark5.4) and this contradicts Proposition 4.4. (cid:3) c x s c Figure 6.
Proof of Lemma 5.8.
ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 31
We complete the proof of Theorem 5.6 by showing that if geodesics joiningpoints of centers or stables do not remain boundedly close to the respective curvesin leaves, then one can construct arcs with shadows with large visual measure:
Lemma 5.9.
Let F be an (cid:82) -covered uniform foliation with hyperbolic leaves ofa closed 3-manifold M and let T be a one dimensional branching subfoliation of F . Assume that for every n ą there is a segment (cid:96) n of a leaf of T such that thegeodesic segment joining the endpoints of (cid:96) n is not contained in the ball of radius n of the segment (cid:96) n . Then, T does not have small visual measure in F .Proof. Just consider the segments (cid:96) n Ă L n and the corresponding geodesic seg-ment r n Ă L n joining the endpoints. By assumption, we know that thereis a point x n P r n at distance larger than n from (cid:96) n , or equivalently, that B L n p x n , n q X (cid:96) n “ H .Since the shadow of (cid:96) n from x n is connected and (cid:96) n intersects both sides of r n we know that the shadow of (cid:96) n through x n has at least half of the visual measurefrom the point x n while it is completely outside the ball of radius n around x n .This implies that T cannot have small visual measure in F . (cid:3) Now we can complete the proof of Theorem 5.6.
Proof of Theorem 5.6.
The statement about visual measure in the case of W s fol-lows by appliyng Proposition 4.11 using T as the stable foliation.. The statementfollows because strong stable leaves cannot be coarsely expanding under P , if P “ γ m ˝ ˆ f n with n ą W c we again apply Proposition 4.11using T as W c . If centers did not have small visual measure in W cs , it followsthat there is a regular pA-pair p ˆ f , γ q associated to p f, W cs q and we can find twodisjoint leaves c , c P Ă W c contained in a leaf L which are fixed by P as well as c , c . Now, Lemma 5.8 gives a contradiction.The statement about arcs or segments of the leaves in the foliations followsfrom Lemma 5.9. The statement about segments is strictly contained in thatLemma. To get the result for rays it is enough to approximate the ray by longerand longer segments which all have the same property. (cid:3) Impossible configurations.
We show that some configurations of the foli-ations in leaves of Ą W cs (or Ą W cu ) are impossible and this will be used to show thatthe leaf space of Ă W c is Hausdorff inside leaves of Ą W cs . The next proposition willcombine well with Lemma 5.8 (which together with Proposition 4.8 gives otherimpossible configurations). We note that the next result works for a single pApair with certain properties and does not need to have the full set of pA pairsthat will be used in next section. In fact, we will need to deal with a case slightlymore general than a pA pair which is when there are only two fixed points in S univ one super attracting and one super repelling. Proposition 5.10.
Let f : M Ñ M be a partially hyperbolic diffeomorphism pre-serving a branching foliation W cs which is (cid:82) -covered and uniform with hyperbolicleaves. Suppose that p f, W cs q admits an admissible regular pseudo-Anosov pair.Let p ˆ f , γ q be a good pair pair and P “ γ m ˝ ˆ f n with n ą so that P has fixedpoints in S univ and such that all fixed points are either super attracting or superrepelling. Let L P Ą W cs be a leaf fixed by P . If there is a center curve c in Ă W c X L with endpoints c ` and c ´ in S p L q such that c ` “ c ´ then one must have that Θ ´ L p c ` q cannot be an attracting fixed point of P . Proof.
We stress that we do not assume that p ˆ f , γ q is a pA-pair. In particular P may have only two fixed points in S univ .Since P p L q “ L we can reduce the proof to ˆ L “ L Y S p L q . The map P inducesa homeomorphism of ˆ L . A point ξ in S univ is a fixed point, attracting or repellingpoint of P if and only if Θ L p ξ q is a fixed point, attracting or repelling point of P in ˆ L . We will prove the result by contradiction assuming that c ` “ c ´ “ a is anattracting fixed point of P in ˆ L . We denote by D p c q to the connected componentof L z c whose closure in ˆ L intersects S p L q only in a .Note that such a center cannot be fixed by P . If it were the case, then itwould be coarsely expanding by Proposition 3.7 and therefore there would bea fixed point x P c by P . Let s p x q be the stable leaf through x . The ray of s p x q intersecting D p c q must land in a “ c ` “ c ´ since a strong stable cannotintersect a center curve twice (cf. Remark 5.5) and therefore the ray is completelycontained in D p c q and lands in c ` . That forces that stable curve to be coarselyexpanding by Proposition 3.7 which is impossible. Compare with Lemma 5.8. Infact the same argument shows that this center cannot be periodic under P aswell.Up to taking the square of P we assume that P preserves orientation whenacting on L , and hence also on S p L q .Consider now the iterates c k : “ P k p c q with k P (cid:90) . Denote by D p c k q “ P k p D p c qq which is the connected component of L z c k whose closure in ˆ L intersects S p L q only in a “ c ` “ c ´ .Consider D “ Ť k D p c k q . Note that D is a P invariant, closed set. Let C bethe set of center leaves which make up the boundary of D .In order to prove the proposition we establish some general claims. The firstone is the place where we use that a is attracting for P . If it were repelling therewould be no a priori contradiction . Claim 5.11.
There cannot be a fixed point of P in D .Proof. Let x P D be fixed by P . If x P D p c k q for some k it follows that oneof the rays of s p x q has to land in c ` which is a contradiction. Otherwise, x isaccumulated by the curves c k , therefore, for large enough k we have that oneray of s p x q intersects c k and therefore enters in D p c k q and must land in c ` , acontradiction. This completes the proof. (cid:3) To continue the proof of Proposition 5.10 we distinguish two options (see Fig-ure 7):(1) The sets t D p c k qqu are not pairwise disjoint.(2) The sets t D p c k qqu are pairwise disjoint.In option (1) there is i ą D p c i q intersects D p c q , hence either D p c i q Ă D p c q or D p c q Ă D p c i q . Hence up to taking a further positive iterate of P weassume that D p c q Ă D p c q or D p c q Ă D p c q . Since it is a positive iterate, thepoint a is still super attracting for P . Option (1)
We assume first that we are in option (1). This situation is by far the harderto deal with. The overall strategy in this case is the following: we find a stableleaf s intersecting D which is fixed by P . This leaf s has one ideal point in a and this will contradict that a is attracting for P . To find such s we essentially Indeed, this behavior can happen for the strong stable/unstable foliations of some partiallyhyperbolic diffeomorphisms such as the ones constructed in [BGHP].
ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 33 D p c q D p c q D p c q D p c q Figure 7.
Option (1) to the left and option (2) to the right. consider the set of all stables intersecting C plus the stable leaves “in between”.We show that this set has a linear order, is invariant by P and P fixes a leaf inthis set. The last step is the hardest and depends on understanding the structureof the boundary of D , how it interacts with the fixed points of P in S p L q . Noticethat a priori the center foliation in L can be very complicated, so there are manytheoretical configurations. Therefore, the first goal is to obtain some useful properties about the boundaryof D . There are similar properties in option (2), but not the same, and option(2) is much easier to deal with. Claim 5.12.
The boundary B D of D in L is a non empty set saturated by centercurves. Every point in B D belongs to a center curve which is a limit of subintervalsof the curves c k . Moreover, the collection C of center leaves are pairwise nonseparated in the center leaf space in L .Proof. Assume that D “ L , then we have that there are compact arcs converginguniformly to some interval in S p L q in the topology of ˆ L “ L Y S p L q , these arcswould have large visual measure and escape to infinity contradicting Theorem5.6. Therefore B D ‰ H .Since D is saturated by center curves, then so is B D . Moreover, if x P B D belongs to a certain center leaf e Ă B D then we have that every compact subin-terval I of e must be accumulated by the sets D p c j q with j Ñ `8 or j Ñ ´8 . This implies that there are arcs I j of c k j converging uniformly to I .Finally let e , e be two distinct center leaves in C . Non separated means thatin the center leaf space they do not have disjoint neighborhoods. As above theleaves e , e are contained in the limit of c j with j Ñ 8 or j Ñ ´8 . Therefore e , e are not separated from each other. (cid:3) Since P preserves the orientation in L the following happens:, if e is in C theneither P p e q “ e or all iterates P n p e q are pairwise disjoint. We can also show: Claim 5.13.
Let e P C a center curve in B D such that P p e q ‰ e . Then t P n p e qu cannot accumulate in a point in L when n Ñ 8 or n Ñ ´8 . The eventual goal, done in Section 6 is to prove that the center foliation in L is actuallyfairly simple, that is, its leaf space is Hausdorff and homeomorphic to (cid:82) . Proof.
Consider G e to be the connected component of L z e which is disjoint from P p e q . Since the curves in C are pairwise non-separated, and P preserves ori-entation in L , we know that P n p G e q are all disjoint. Assuming that t P n p e qu accumulates in some point x P L with n Ñ ˘8 , we can fix a local product struc-ture neighborhood around x for the center foliation and we can see t P n p G e qu accumulating in this point. Since these sets are all disjoint, then this means thatthe leaves t P n p e qu accumulate on a local product structure box in more than oneconnected component, which is impossible. (cid:3) Claim 5.14.
Let e be a leaf in C . If e has an ideal point ξ which is a fixedpoint of P then the other ideal point ν of e is different from ξ and one of them isattracting and one repelling. In addition the ideal points of e cannot be in distinctcomplementary components of the set of fixed points of P in S p L q .Proof. Suppose e is a leaf in C which has an ideal point ξ fixed by P . Let ν bethe the other ideal point of e . Suppose first that ν is distinct from ξ . Then since P p e q is non separated from e and c k converges to both e and P p e q it follows that P p e q “ e . If both endpoints of e are either attracting or repelling for P thenthere is a fixed point of P in e , hence a fixed point of P in D , disallowed in Claim5.11. So one of the ideal points of e is attracting and the other one is repelling.Suppose now that ν “ ξ . Let G e be the component of L ´ e which accumulatesonly in ξ in S p L q . Since ξ must be either super attracting or super repelling, wededuce that P p e q ‰ e and therefore the iterates P i p G e q are all distinct. Moreover,by Claim 5.13 we know that P i p e q cannot accumulate on a point x P L whichimplies that the sets P i p G e q converge as i Ñ ˘8 to ξ . However, since ξ is superattracting or super repelling it follows that P i p e q cannot converge to ξ as i Ñ ´8 or i Ñ `8 . This proves the first assertion of the claim.Finally suppose that e has ideal points in two distinct complementary com-ponents of fixed points of P in S p L q . Up to an iterate these complementarycomponents are fixed by P . Then since e is a boundary leaf of D this impliesthat P p e q “ e . In particular the ideal points of e are fixed by P and are notin complementary of the set of fixed points of P . This finishes the proof of theclaim. (cid:3) We now define a set S of stable leaves which will produce a P invariant stableleaf intersecting D . The construction of S is geometric and not dynamical. Forsimplicity assume that c j converges to C when j Ñ 8 . The case when c j convergesto C when j Ñ ´8 is entirely analogous and we address that later.Let S be the set of stable leaves s in L so that there is j P (cid:90) so that s intersects c j for any j ě j . Each such stable leaf s intersects some c j . Since c j has bothideal points equal to a , it follow that s has a ray limiting on a “ c ` “ c ´ . Eachstable leaf s intersecting C intersects c j for all j ě j (the j depends on s ), so s is in S . In addition if s , s are in S then they intersect c j for all j ě j forsome j (take a j that works for both). For any stable s intersecting c j between s X c j and s X c j then s intersects c j for any j ě j , so s is also in S . This isbecause c j , c j and s , s form a “quadrilateral” in L and s intersects c j , henceintersects c j also. It follows that s is in S . Therefore the set S is linearly ordered.Since the subset of S between s and s is order isomorphic to an interval then S is order isomorphic to the reals. With the quotient topology it is homeomorphicto the reals. Since we took a square of P if necessary, then the map P preservesthis order.If on the other hand c j converges to C when j Ñ ´8 , then in the definition of S we require a j so that s intersects all c j for j ď j . ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 35
Let
I, J be the connected components of S p L q minus the set of fixed pointsof P in S p L q , so that I, J have one endpoint equal to a . We now define a subset S I of S associated with I . The definition will depend of whether there is a leafof C with an ideal point in I or not. Consider first the case that there is a leaf e in C with an ideal point in I . By Claim 5.14 the other ideal point of e is alsoin I . Let A i be the set of stable leaves in L intersecting P i p e q . Let S I be thesmallest connected set of S containing all A i . In this case for any s in S I then s has an ideal point in I . In fact for any s in S it has an ideal point in a . If theother ideal point x of s is in I then x is in the closed segment contained in I withendpoints P i p z q and P i ` p z q where z is an ideal point of a stable intersecting e and i is some integer. Hence s separates two elements in S and intersects c j forall j ě j (for some j ) hence s is in S . It follows that in this case S I is exactlythe set of stables intersecting D which have one ideal point in I . In particularthe definition of S I is independent of the particular leaf e that we start with.The other possibility is that there is no leaf of C with an ideal point in I . Wedeal with this case now. Let x in I . We claim that there is a neighborhood V of x in L Y S p L q so that V X L is disjoint from D . Suppose not. If for somesuch V we have that V X L Ă D , then we get a sequence of arcs in c k (with c k limiting to C as k Ñ 8 ) so that they escape compact sets in L and limitto V X S p L q . These arcs do not have small visual measure, violating Theorem5.6. So this cannot happen. Choose V i with i P N a basis neighborhood of x in L Y S p L q , with V i X S p L q always contained in I . By assumption, for each i thereis a point y i in V i X c k i for a suitable choice of k i . If the k i can be chosen constantequal to k then c k has one ideal point in V k X S p L q . But this is impossible byhypothesis as V k X S p L q Ă I . So up to subsequence we can assume that k i arepairwise distinct. Since the c k i have both ideal points outside of I then eitherthey escape compact sets, contradicting Theorem 5.6, or up to subsequence theykeep intersecting a fixed compact set. This is impossible since the elements in C are pairwise non separated from each other, cf. Claim 5.12. This shows thatthere is such V as above, so that p V X L q X D “ H . In this case let e be theunique leaf of C which separates V X L from the interior of D . By P invarianceof D and the fact that no ideal point of e is in I , it follows that P p e q “ e . Inaddition one ideal point of e is a . This is because e separates V X L from theinterior of D and the interior of D has points limiting to a . Since e is fixed by P the other ideal point of e is a repelling fixed point of P by Claim 5.14, and henceit is not a . Let now I ex be the open interval of S p L q determined by the idealpoints of e and which contains I . We remark that I ex is disjoint from J . In thiscase let S I be the set of stable leaves intersecting e .Notice that S I is again a connected subset of S . In either case we remark thatif s is a leaf in S I then no ideal point of s is in J .Notice that in either case S I is P invariant. In the same way we define a set S J . Claim 5.15.
The sets S I , S J are disjoint.Proof. Suppose that there is a leaf e in C with an ideal point in either I or J .For simplicity assume an ideal point in I . Then by construction for every leaf s in S I it has an ideal point in I . Since no leaf in S J has an ideal point in I , theclaim is proved in this case.The remaining case is that we have the intervals I ex and J ex , which are definedby leaves e, e in C . In this case S I is the set of stable leaves intersecting e and S J is the set of stable leaves intersecting e . Since e, e are distinct but non separated from each other, no stable leaf can intersect both of them. Hence again S I X S J “ H . This proves the claim. (cid:3) Since S I , S J are disjoint, let s be the stable leaf in S corresponding to theendpoint of S I separating it from S J in S . Then since both S I , S J are fixed by P , so is s . Since s is in S then it intersects c j for some j and hence has an idealpoint a . This contradicts the fact that a is an attracting fixed point of P .This finally finishes the proof of Proposition 5.10 assuming option (1). Option (2)
We now assume option (2).
Claim 5.16.
In option (2) we have that as k Ñ ˘8 then D p c k q can only accu-mulate in a “ c ` “ c ´ .Proof. In option (2) the sets D p c k q are pairwise disjoint. An argument entirelyanalogous to that of Claim 5.13 shows that D p c k q cannot accumulate anywherein L as k Ñ 8 or k Ñ ´8 .If the collection D p c k q accumulates in another point of S p L q besides a , thensince it does not accumulate in L it will have subsegments which limit uniformlyon non empty intervals of S p L q . In particular these segments escape compact setsin L . This violates that the center foliation has small visual measure, Theorem5.6. This finishes the proof of the claim. (cid:3) Now we can complete the proof in option (2). As k Ñ ˘8 , the D p c k q cannotaccumulate in L or in any other point of S p L q besides a . Choose a neighborhood U of a which is contracting under P as in Proposition 3.7. Choose U sufficientlysmall so that D is not contained in U . Then for k big negative D p c k q is containedin U , and applying P ´ k sends D p c k q inside of U , but also to D p c q not containedin U , contradiction. This completes the proof of Proposition 5.10. (cid:3) Hausdorff center leaf space
In this section we show that under some assumptions the center leaf space of Ă W c has to be Hausdorff in leaves of Ą W cs . This is an important step in the proofof our main theorems and will use all the results on pseudo-Anosov pairs we havebeen developing so far. We use the abbreviation pA pairs for pseudo-Anosovpairs.To be able to exclude non-Hausdorff leaf space we will need enough pA pairsto be able to force certain configurations (see Remark 6.3 below). This will bedefined precisely in § § § Theorem 6.1.
Let f : M Ñ M be a partially hyperbolic diffeomorphism pre-serving branching foliations W cs and W cu . Assume that p f, W cs q has full pseudo-Anosov behavior (cf. Definition 6.7). Assume also that W cu is (cid:82) covered. Then,inside each leaf of Ą W cs , the foliation Ă W c by center curves has leaf space which isHausdorff. Recall that a one dimensional (branching) foliation r T in a complete simplyconnected surface L has Hausdorff leaf space if for every pair of curves of r T in L it follows that the positive (closed) half space in L determined by one of thecurves is contained in the positive (closed) half space determined by the other. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 37
Remark . Definition 6.7 is quite restrictive and asks for the existence of severalpseudo-Anosov pairs for f . We suspect that the only examples which verify theseassumptions are the ones we treat in Theorem B and Theorem C. We note howeverthat we do not ask that the actions of the good pairs in the universal circle of W cs and W cu coincide (this is immediate in the context of Theorem C, but not apriori obvious for Theorem B). Remark . Until now, all arguments used a given pA pair and then found se-quences of curves that approached the universal circle in certain configurationsthat would ensure that some of their points belong to the basins of attract-ing/repulsion of the fixed points in S univ of the pA pairs. In this section thestrategy will be different. We will fix a curve and find sequences of pA pairswhose configurations will force that the curve has some points in basins of at-traction of different fixed points of the pA pairs (as the configuration required inProposition 4.8). Two delicate issues with this appear: ‚ the curves we will consider already have limit points and approach theboundary very fast (cf. Theorem 5.6) so we need that the configurationof attracting/repelling points of the pA pairs are very special; ‚ also, the core T P of a pA pair depends somewhat on the particular pApair we choose, that is why it will be important to consider pA pairswhich are related to the same object in M (i.e. different lifts of the same‘orbit’) so that we get some uniform estimates.For these reasons, we will need to restrict to a class of diffeomorphisms thatwe will later show contains the classes we are studying in this paper to showTheorems B and C..6.1. Diffeomorphisms with a full set of pseudo-Anosov pairs.
In somearguments we will need not only one pseudo-Anosov pair but that its conjugates(cf. Definition 3.18) fill the universal circle in a particular way.
Remark . In what follows one should have in mind the difference betweena pseudo-Anosov diffeomorphism of a surface and a reducible diffeomorphismof a surface with a pseudo-Anosov piece. One can also think about regulatingflows for uniform foliations in atoroidal manifold versus manifolds with atoroidalpieces but non-trivial JSJ decomposition. (Recall Examples 3.3 and 3.4.) Whenthere is a unique pseudo-Anosov piece, the laminations are minimal, so every(regular) periodic orbit verifies that its stable/unstable manifold is dense in thestable/unstable lamination of the pseudo-Anosov map which forms a “full lami-nation” (see [Ca ] and references therein).We consider a pair p f, F q with the periodic commuting property (cf. Definition4.5) and let p ˆ f , γ q be an admissible regular pA-pair with attracting points a , a and repelling points r , r in S univ (here the action is with respect to a lift P “ γ m ˝ ˆ f k as in Notation 4.6). Definition 6.5.
The regular pA-pair p ˆ f , γ q is a full pair (for P “ ˆ f k ˝ γ m ), ifthere are α ą d ą η withstarting point x in a leaf L P r F there is: ‚ a compact interval I in the leaf space of r F , ‚ and for every n ą
0, a deck transformation β n P π p M q such that β p L q P I so that:if we denote by g an the geodesic in L joining Θ L p β n a q with Θ L p β n a q and g rn thegeodesic in L joining Θ L p β n r q with Θ L p β n r q then: ‚ either g an or g rn intersects η in a point x n making angle larger than α , ‚ d L p x , x n q ą n and d L p x n , g X g q ă d .We will use this property to obtain the following important result. Proposition 6.6.
Let p f, F q have the periodic commuting property and admittinga full pair p ˆ f , γ q . Then, given a geodesic ray η in a leaf L P r F from a point x P L with ideal point Θ L p ξ q ( ξ P S univ ) there exists a conjugate pair p ˆ f , γ q ´ with P the corresponding conjugate of P (cf. Definition 3.18) such that either x and ξ belong to different basins of attraction of the fixed points of P in S univ or theybelong to different basins of repulsion of the fixed points of P .Proof. Consider the geodesic ray η in L with starting x P L and ideal pointΘ L p ξ q . Let a , a be the attracting points of P . Since p ˆ f , γ q is a full pair,without loss of generality we can assume that there is a sequence γ n P π p M q such that the geodesic g n in L with ideal points Θ L p γ n a q and Θ L p γ n a q makesangle larger than α with η and intersects η in a point x n at distance larger than n from x . In addition if h n is the geodesic in L with ideal points Θ L p γ n r q andΘ L p γ n r q then d L p g n X h n , x n q ă d .Let P “ γ n ˝ P ˝ γ ´ n .Extend η in L beyond x to a full geodesic still denoted by η and with otherideal point Θ L p ν q . Now we map back by γ ´ n . The fact that γ n p L q is in a compactinterval of the leaf space means that the slithering distance of γ n is bounded [Th ]and so is the slithering distance of γ ´ n . In other words γ ´ n p L q is in a compactinterval, which we denote by J .Suppose that up to subsequence that one of γ ´ n p ξ q or γ ´ n p ν q converges to a or a . Without loss of generality assume that γ ´ n p ξ q converges to a or a . Upto another subsequence assume that γ ´ n p L q converges to L . Notice that g n X h n is a globally bounded distance from T P X L ´ because of the following:(i) T P { ă γ ą is compact, and(ii) Since deck transformations are isometries of Ă M they induce metrics inthe quotients, and also T P { ă γ ą , T P { ă γ n ˝ γ ˝ γ ´ n ą are isometric.Since d L p g n X h n , x n q is bounded by d , it follows that d γ ´ n p L q p γ ´ n p x n q , T P X γ ´ n p L qq is bounded above, so we can assume γ ´ L p x n q also converges to y . The sequenceof geodesics γ ´ n p g n q in γ ´ n p L q have ideal points Θ a , Θ a , so converges to thegeodesic g in L with ideal points Θ L p a q , Θ L p a q . In addition the sequence γ ´ n p η q converges to a geodesic in L through y and making an angle with g atleast α . This is impossible since γ ´ n p ξ q converges to either a or a . Thereforenone of γ ´ n p ξ q , γ ´ n p ν q converges to either a or a .By hypothesis ξ, ν P S univ are in distinct components of S univ t a , a u . Theprevious paragraphs show that there are fixed interval neighborhoods I, J of therepellers r , r not containing either a , a in their closures, so that γ ´ n p ξ q , γ ´ n p ν q are always in I, J respectively for n sufficiently big.. Let U I , U J be neighbhor-hoods of r , r respectively as constructed in Proposition 3.7. Then γ ´ n p ξ q is in U I for n big. Since d L p x , x n q ą n , then eventually γ ´ n p x q is in U J .This shows that ξ, x belong to the basins of repulsion of γ n p r q , γ n p r q respec-tively under P . ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 39
This proves the proposition. (cid:3)
Now we are ready to give the definition we will use to get Theorem 6.1.
Definition 6.7.
We say that p f, F q has full pseudo-Anosov behavior if p f, F q hasthe periodic commuting property (cf. Definition 4.5) and:(i) every admissible good pair (cf Definition 3.17) of p f, F q , up to iterate, hasonly super attracting and super repelling fixed points. and,(ii) it contains a regular pA-pair which is a full pair (cf. Definition 6.5).6.2. Distinct landing points.
We first need the following auxiliary result. Thisis the place where we will use the full pseudo-Anosov behavior on one foliation.
Proposition 6.8.
Let f : M Ñ M be a partially hyperbolic diffeomorphismsuch that it preserves branching foliations W cs and W cu and such that p f, W cs q has full pseudo-Anosov behavior. Assume also that W cu is (cid:82) -covered. Then, forevery c P Ă W c X L with L P Ą W cs we have that the endpoints c ` and c ´ of c in S p L q are different. The proof of this statement will require to first iterate in W cs until we get acenter curve both of whose endpoints land in a single fixed point of a pA pair for p f, W cs q . If the fixed point is super attracting one can apply Proposition 5.10 toget a contradiction. If the fixed point is repelling, we need to use Theorem 5.6and an analysis of the center unstable foliation W cu to derive a contradiction. Lemma 6.9.
Let f be as in Proposition 6.8 and assume there is a center curve c P Ă W c X L for L P Ą W cs so that c ´ “ c ` . Then, there exists a regular pA-pair p ˆ f , γ q such that P as in Notation 4.6 fixes a leaf L P Ą W cs which has a centercurve c so that both endpoints coincide and so that c is fixed by P .Proof. Pick a point x P c and consider the geodesic ray η from x to Θ L p ξ q “ c ´ “ c ` . Since p f, W cs q has full pseudo-Anosov behavior it has an admissible fullpair p ˆ f , γ q which is a regular pA pair.We will use Proposition 6.6. By Proposition 6.6 we deduce that, up to con-jugating p ˆ f , γ q , there is a P so that x , ξ are in different basis of attraction ofeither P or P ´ .We will apply Proposition 4.8 with T “ W c the center foliation.We consider the case where the point x P c is in the basin of repulsion of arepeller of P and ξ in the basin of repulsion of the other repeller of P . ApplyingProposition 4.8 to P ´ we obtain that iterating by P ´ n the leaf L converges toa leaf L fixed by P and P fixing disjoint center curves c and c which join therepelling points. (The other case is symmetric and obtains curves that join theattracting points of P by iterating forward.)By Addendum 4.9 one can also see that there is a center curve c in L between c and c and so that both endpoints of c in L are equal. Since the curve isbetween c and c it follows that its forward iterates remain in a compact region,and so the curves converge to at least one curve c whose endpoints coincide withthe endpoints of c and which is fixed by P .The symmetric case is dealt with using P ´ instead of P . (cid:3) Proof of Proposition 6.8.
Let c given by the previous lemma. By Proposition5.10 we get that the endpoints of c must correspond to a super repelling pointof P . Since c is fixed by P , there is a leaf F P Ą W cu containing c and fixed by P .Since the action of P on c is coarsely contracting, there is a fixed point x P c by P . Let c , c be the rays of c zt x u .We first consider the situation in L and the foliation W cs . Both c , c limit tothe same point z “ Θ L p ξ q in S p L q with ξ super repelling for P . Theorem 5.6implies that the geodesic ray in L starting in x and with ideal point z is containedin uniform neighborhoods in L of c and c . In particular, there are sequences p n P c and q n P c converging to z in L Y S p L q so that d L p p n , q n q is bounded.It follows that d Ă M p p n , q n q is bounded.We now look at the center unstable foliation W cu . Since W cu is (cid:82) -covered, thisimplies that in F the points p n and q n are a bounded distance apart. This isbecause F is uniformly properly embedded in Ă M . Let (cid:96) n be the geodesic segmentin F from p n to q n . By trimming (cid:96) n or replacing p n , q n if necessary, we assumethat (cid:96) n intersects c only in p n , q n , still keeping the length of (cid:96) n globally bounded.Since ξ is super repelling for P (acting on the universal circle of W cs ) it followsthat d Ă M p p n , P p p n qq converges to infinity and likewise for q n . The length of P p (cid:96) n q is uniformly bounded. In particular for n big enough P p (cid:96) n q is disjoint from (cid:96) n .Fix one such n . Let D be disk in F bounded by (cid:96) n and the segment in c from p n to q n . By the above P p D q is strictly contained in D . There is a ray of theunstable leaf of x entering D . This ray intersects B D . This ray is expanded by P . This contradicts that P p D q is a subset of D .This finishes the proof of Proposition 6.8. (cid:3) Proof of Theorem 6.1.
The proof of Theorem 6.1 is very similar to theprevious argument.Assume by contradiction that there is a leaf L P Ą W cs on which the leaf space of Ă W c is non Hausdorff. Consider two center leaves c, c P L which are non-separatedin the sense that there is a sequence c n of center leaves such that c n convergesboth to c and c (it may converge to other center leaves too). Up to changingorientation of the center foliation, we can assume that there are arcs I n of c n which approximate the points c ` and p c q ´ in ˆ L “ L Y S p L q which may ormay not coincide.By Proposition 6.8 c ´ ‰ c ` . Hence by Proposition 5.10 we can choose as inLemma 6.9 an admissible regular pA pair p ˆ f , γ q which verifies that in L eitherthe geodesic joining the attracting points or the repelling points separates c ` from the ideal points of the center curves c n . As before, we assume that it isthe geodesic joining the attracting ideal points that makes the separation (as theother case is symmetric ).We can assume by further conjugating p ˆ f , γ q that it verifies that both idealpoints of the curves c n belong to the same basin of repulsion of the repeller r of P . Since c ` belongs to the basin of repulsion of the other repeller r it followsthat for n large enough, the segment I n intersects the basin of repulsion of r .We can then apply Proposition 4.8 to P k , k ă L P r F such that it contains two center curves which are disjoint and join therepelling points of P . Moreover, between these center curves there is a fixedcenter curve both of whose points coincide by Addendum 4.9. This contradictsProposition 5.10 and concludes the proof of Theorem 6.1. By this we mean that the Haudsorff limit of the arcs I n in ˆ L contains both c ` and p c q ´ . In Proposition 5.10 the case where the configuration allowed to iterate forward was slightlysimpler since the argument was not symmetric, but since here we are going to reduce to thatcase, here it is symmetric.
ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 41 Quasigeodesic behavior
In this section we want to show that under our assumptions the centers behaveas uniform quasigeodesics in leaves of center stable and center unstable branchingfoliations.Recall that for k ą
1, an embedded rectifiable curve (cid:96) Ă L in a completeRiemannian manifold is called a k - quasigeodesic if one has that for all x, y P (cid:96)d (cid:96) p x, y q ď kd L p x, y q ` k. where d L denotes the Riemannian distance in L and d (cid:96) is the length along (cid:96) .Note that d (cid:96) p x, y q ě d L p x, y q always. Definition 7.1.
Let F be a (branching) foliation in a closed 3-manifold M . Wesay that a one dimensional branching subfoliation T of F is by uniform quasi-geodesics if there exists k such that every curve (cid:96) of r T in L P r F with the inducedpath metric is a k -quasigeodesic. Remark . The fact that a subfoliation is by uniform quasigeodesics is inde-pendent of the metric in M since M is compact. Only the constant may change.In our setting we typically work with (branching) foliations whose induced met-ric is negatively curved, where quasigeodesics have very meaningful geometricproperties thanks to the classical Morse lemma (see [BH, § III.H.1]).Now we can state the main result of this section:
Theorem 7.3.
Let f : M Ñ M be a partially hyperbolic diffeomorphism pre-serving branching foliations W cs and W cu such that both p f, W cs q and p f, W cu q have full pseudo Anosov behavior (Definition 6.7). Suppose that there is p ˆ f , γ q aregular full pair for p f, W cs q which is also a good pair for p f, W cu q . Then W c isby uniform quasigeodesics in both W cs and W cu . Note that uniform quasigeodesics in leaves whose metric vary continuously canbe followed in nearby leaves, so we deduce that:
Corollary 7.4.
The endpoint maps c ÞÑ c ˘ from the leaf space L c of the center fo-liation Ă W c to S univ is continuous and π p M q -equivariant. It is also ˆ f -equivariantfor ˆ f a lift of f to Ă M (see Proposition 2.5). We explain what we mean by c ˘ . Suppose that c is contained in a leaf L of Ą W cs .Let b be the ideal point of c in the positive center direction. Then c ` “ Θ ´ L p b q .Similarly for c ´ .7.1. Tracking geodesics.
Consider T a one-dimensional sub-branching foliationof a branching foliation F of M . We assume that F is (cid:82) -covered and uniform andby hyperbolic leaves so that we can apply all what was developed in § § r F of Ă M whichis subfoliated by a (branching) foliation r T and we choose an orientation for both.We will say that T has efficient behavior in F if the following conditions hold:(i) the leaf space of r T in each leaf L P r F is Hausdorff,(ii) each curve (cid:96) P r T in a leaf L P r F has well defined limit points (cid:96) ´ and (cid:96) ` in S p L q which are different(iii) there is R ą (cid:96) P r T and x P (cid:96) Ă L P r F if we denoteby r ˘ x the geodesic ray in L joining x with (cid:96) ˘ then we have that r ˘ x iscontained in the R -neighborhood in L of the ray of (cid:96) from x to (cid:96) ˘ .(iv) T has small visual measure (cf. Definition 4.10). Remark . In the previous sections we have established that if f : M Ñ M is a partially hyperbolic diffeomorphism in the hypothesis of Theorem 7.3 thenthe center (branching) foliation W c has efficient behavior in both W cs and W cu :Point ( i ) is done in §
6, point ( ii ) in § iii ), ( iv )in Theorem 5.6.The following will be an auxiliary result to show the quasigeodesic behavior. Lemma 7.6.
Let T having efficient behavior in F and let c n P r T be a sequenceof leaves. Assume that c n Ă L n P r F so that L n Ñ L , c n Ñ c P L and there arepoints x n P c n so that x n Ñ x P c . Assume there is a point y n in the ray of c n zt x n u with positive orientation so that y n Ñ ξ P S univ (as in Definition 3.6).Then, the endpoint c ` of the positively oriented ray of c zt x u is ξ is Θ L p ξ q .Proof. Suppose this is not true, let c n , x n , y n failing this condition. Up to sub-sequence assume that y n converges to ξ , with c ` “ Θ L p ξ q . Let ν P S univ withΘ L p ν q “ c ` .Since c ` “ Θ L p ν q and c n converges to c in the center leaf space, then c n also haspoints z n between x n and y n so that z n converges to Θ L p ν q . Consider the segments J n in c n from z n to y n . These segments do not have visual measure convergingto zero as n Ñ 8 , because Θ L p ν q “ Θ L p ξ q . Since T has small visual measureit follows that these segments cannot escape compact sets Ă M and converge to acollection of center leaves in L . Let (cid:96) be such a center leaf. In particular there are w n in J n converging to w in (cid:96) . If c “ (cid:96) , then the the local product structure offoliations (in the center leaf space) shows that the length of segments in c n from x n to w n is bounded, so the length in c n from x n to z n would also be boundedcontradiction.We conclude that (cid:96), c are distinct center leaves in L . By Theorem 6.1 thecenter leaf space restricted to L is Hausdorff. Hence there is a transversal to thecenter foliation in L from x to w . This transversal produces in nearby leaves L n ,transversals to the center foliation in L n from x n to w n . This is a contradiction,because x n and w n are in the same center leaf in L n . This finishes the proof. (cid:3) Clearly the same statement holds for the negatively oriented ray.
Remark . We say that the endpoints of curves in T vary continuously if givena sequence of leaves (cid:96) n P r T , (cid:96) n in L n leaves of Ą W cs , with endpoints (cid:96) ` n and (cid:96) ´ n in S p L n q and so that (cid:96) n converges to (cid:96) , in the leaf space of r T and (cid:96) Ă L , where L is the limit of L n , then the sequencesΘ ´ L n p (cid:96) ` n q , Θ ´ L n p (cid:96) ´ n q in S univ converge to Θ ´ L p (cid:96) ` q and Θ ´ L p (cid:96) ´ q respectively. It is worth making the remarkthat if the sequence of leaves (cid:96) n converges to (cid:96) in the leaf space of r T it meansthat given a compact part of (cid:96) it will be well approached by the curves (cid:96) n . Theproof of the previous Lemma can be adapted to show that the endpoints of thecurves in T vary continuously. Since we will only use the statement above, andcontinuity also follows a posteriori from the fact that T is uniformly quasigeodesic(cf. Corollary 7.4), we do not prove this here. If we consider a sequence of points x n Ñ x one may choose leaves of r T through x n whichare far from a given curve passing through x due to potential non unique integrability of the(branching) foliation. Convergence in the leaf space is needed to make this work. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 43
Now we can use the previous lemma to show the following property whichimplies one of the main consequences of being uniform quasigeodesic. This willallow us to show the quasigeodesic property in the next subsection.
Lemma 7.8. If T has efficient behavior in F then there is R ą such that forevery center leaf (cid:96) P r T contained in L leaf of Ą W cs , then the geodesic g in L withideal points (cid:96) ` and (cid:96) ´ is at Hausdorff distance less than R in L from (cid:96) . Thesame results holds for segments or rays in leaves of r T .Proof. Notice that condition (ii) of a efficient lamination says that (cid:96) ´ “ (cid:96) ` , sothe geodesic g is defined.We first prove this for finite segments: there is a uniform constant R ą L P r F , the Hausdorff distance in L between a geodesic segment in L joining the endpoints of an arc I Ă (cid:96) P r T X L and I is less than R . Assumethat this is not the case. Then, we can find a sequence I n of segments of leaves (cid:96) n P r T , (cid:96) n Ă L n P r F , so that there is a point x n P I n at distance larger than n in L n from the geodesic segment g n in L n joining the endpoints of I n .Up to composing with deck transformations we can assume that the points x n belong to a fixed compact fundamental domain of M in Ă M and therefore, upto subsequence, that x n Ñ x P Ă M , that (cid:96) n Ñ (cid:96) through x , and that L n Ñ L containing (cid:96) . Up to another subsequence assume that one of the endpoints of g n converges to a point ξ in S univ .Since the distance in L n from x n to g n converges to infinity, and g n are geodesicsegments in L n it follows that visual measure of g n in L n from x n converges to0. In other words both endpoints of g n in L n converge to the same point ξ of S univ . Applying Lemma 7.6 it follows that both endpoints of (cid:96) are Θ L p ξ q . Thiscontradicts condition (ii) of a efficient lamination. This proves the result forsegments.To get the result for full center leaves, take x P (cid:96) and consider a sequence I n ofintervals of (cid:96) from z n to a point y n so that y n Ñ (cid:96) ` and z Ñ (cid:96) ´ . Since y n Ñ (cid:96) ` , z n Ñ (cid:96) ´ in ˆ L “ L Y S p L q it follows that the geodesic segments from z n to y n converge uniformly on compact sets to the geodesic with ideal points (cid:96) ` and (cid:96) ´ .Therefore the result holds maybe by taking R slightly larger. A similar proofholds for rays. (cid:3) The quasigeodesic behavior.
Here we show the following result which isa standard. A similar result in a slightly different setting can be found in [FM].
Proposition 7.9.
Let T be a one dimensional (branching) foliation of M whichsubfoliates F . Assume that there exists R ą such that for every L P r F andevery finite segment I in a leaf (cid:96) P r T X L there is a geodesic segment in L withsame endpoints as I which is at Hausdorff distance in L less than R ´ from (cid:96) .Then T is by uniform quasigeodesics in F .Proof. As seen by the proof of the last Lemma the condition implies that fullleaves of r T have distinct ideal points and are R distant from the correspondinggeodesics in their respective leaves. This also immediately implies that the leafspace of r T in any leaf of r F is Hausdorff.Let a ą R . We first claim that there is a global length a so that if a segment I in a leaf (cid:96) of r T contained in L leaf of r F has length more than a then the distancein L between the endpoints of I is more than a . Otherwise find segments I n oflength ą n with endpoints x n , y n less than a in their respective leaves. Up todeck transformations and subsequences assume that x n Ñ x , y n Ñ y both in L , which is the limit of leaves L n containing I n . The leaves (cid:96) n through x n , y n converge to a leaf b through x and a leaf b through y . This is in the leafspace of r T . If b “ b then the lengths of (cid:96) n between x n and y n are boundedcontradiction. Hence b , b are distinct leaves. There is a trasversal from x to y and this leads to transversals in respective leaves of r F from x n to y n ,contradiction. This proves the claim.Given (cid:96) leaf of r T in leaf L consider the geodesic g in L with same ideal pointsas (cid:96) and the orthogonal projection from (cid:96) to g . Since (cid:96) is in a neighborhood ofsize R in L from g , then the claim above shows that every time we follow along (cid:96) a length ě a the projection to g moves forward at least R . This proves theuniform quasigeodesic behavior of leaves of r T . (cid:3) Proof of Theorem 7.3.
As observed in Remark 7.5 we know that under our as-sumptions the center foliation T has efficient behavior in W cs , W cu . Properties( ii ) and ( iii ) of efficient behavior plus Lemma 7.8 imply that W c is in the hy-pothesis of Proposition 7.9 with respect to both W cs and W cu . The result thenfollows. (cid:3) The collapsed Anosov flow property
In view of the previous section we can deduce:
Theorem 8.1.
Let f : M Ñ M be a partially hyperbolic diffeomorphism preserv-ing branching foliations W cs and W cu which are uniform, (cid:82) -covered, and suchthat both p f, W cs q and p f, W cu q have full pseudo-Anosov behavior. Then, f is acollapsed Anosov flow. This follows directly from Theorem 7.3 and the results from [BFP] (note thatwith the terminology of [BFP] this will give that f is a strong collapsed Anosovflow ). We give here a detailed sketch of a proof of this fact in this simpler settingsince some arguments can be simplified and we will use many of the notions toget uniqueness of branching foliations. Definition 8.2.
A one dimensional branching foliation T in an (cid:82) -covered uniformfoliation F by hyperbolic leaves is said to be a quasigeodesic fan foliation if forevery L P r F there is a point p “ p p L q P S p L q called the funnel point such thatthere is a bijection from the leaf space of r T X L and points in S p L qz p so thatthe leaf of r T X L corresponding to the point q P S p L q is a quasigeodesic joining q and p .The key point of the proof of Theorem 8.1 is to show that the center branchingfoliation W c is a quasigeodesic fan foliation in both W cs and W cu since this allowsto produce a (topological) Anosov flow rather easily (in fact, an expansive flowpreserving transverse foliations, which is equivalent to being a topological Anosovflow). We refer the reader to [BFP] for details on this, we will concentrate herein explaining how to obtain that centers form a quasigeodesic fan foliation justby knowing that the leaves of Ă W c are uniform quasigeodesics in Ą W cs and Ą W cu (cf.Theorem 7.3).To prove this, we follow a path which is somewhat more direct than the onetaken in [BFP] since the (cid:82) -covered property simplifies the arguments.First, we notice that the branching foliations W cs and W cu must be minimal.Recall that being minimal means that there is no closed π p M q invariant set inthe leaf space of the (branching) foliation in the universal cover. See [BFFP ,Appendix F] for more discussion. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 45
Proposition 8.3.
Let f : M Ñ M be a partially hyperbolic diffeomorphismpreserving branching foliations W cs and W cu which are uniform and (cid:82) -coveredand such that the center foliation W c is a quasigeodesic foliation both in W cs and W cu . Then W cs and W cu are minimal.Proof. We argue for W cs since W cu is symmetric. To see this, we use the factthat since they are foliated by quasigeodesics then every leaf of Ą W cs has cyclicstabilizer in π p M q . This way, if there was a proper minimal set, one can constructa solid torus region exactly as in the proof of Proposition 6.1 of [BFFP ] to makea volume versus length argument to reach a contradiction. (cid:3) Proof of Theorem 8.1.
The hypothesis imply that W cs , W cu are transversely ori-entable.One can first argue similar to what is done in [Ca , §
5] to see that the set ofleaves of Ą W cs on which the foliation Ă W c is a (weak)-quasigeodesic fan is a π p M q and ˆ f -invariant closed set of the leaf space of Ą W cs which is non-empty. Thereforethis set is everything because of minimality. By weak quasigeodesic fan we meanthat all center leaves share a common ideal point, but we allow several curves ofthe foliation in a leaf of Ą W cs to joint the same pair of points. In a quasigeodesicfan no two distinct leaves have the same pair of ideal points. One gets the lesspowerful weak quasigeodesic fan property because to apply arguments similar tothose in [Ca , §
5] one needs to tighten up the foliation to an equivariant geodesicfoliation on leaves. The arguments in [Ca ] are for geodesic and not quasigeodesicleaves. After this is one, it is however possible to show that the subset of theleaf space of Ą W cu corresponding to the interval of centers in a Ą W cs leaf that jointhe same pair of points produces an open and π p M q -invariant sublaminationthat cannot be the whole leaf space. So again using minimality we exclude thispossibility. We refer to detailed proofs, which work with much more generality,in Propositions 6.9, 6.15 and 6.20 of [BFP]Once the center (branching) foliation W c is a quasigeodesic fan foliation in W cs and W cu we apply the approximation foliation result (Theorem 2.1) to obtain atrue foliation W cε which subfoliates the approximating foliations W csε and W cuε with the same quasigeodesic fan property. One can then show that this gives anexpansive flow in M which, by virtue of preserving a pair of foliations, is topo-logically Anosov [IM, Pat]. Since the foliation is (cid:82) -covered, the flow is transitiveand therefore orbit equivalent to a true Anosov flow by [Sha]. The maps given bythe approximating foliation allow one to construct the collapsing map h whichmust then be intertwining the action of f with a self orbit equivalence associatedto how it permutes the orbits of the flow. The construction of semiconjugacy h in [BFP] is fairly complex due to the possibility of branching in the foliations.We again refer the reader to [BFP] for detailed arguments which work in muchmore generality. See Theorems B and D of [BFP]. (cid:3) Funnel directions
In this section we obtain a couple of technical properties.Let f be a partially hyperbolic diffeomorphism satisfying the hypothesis ofTheorem 8.1. In particular in every leaf L of either Ą W cs or Ą W cu , the centerfoliation is a fan. The stable funnel direction of a center c in a leaf L of Ą W cs isgiven by the orientation in c towards the funnel point in L . Similarly one definesthe unstable funnel direction. By Corollary 7.4 the stable funnel direction variescontinuously and clearly it is invariant by deck transformations. Notice that the stable funnel direction is defined a priori for points in center leaves contained incenter stable leaves and not just on points. However any two center leaves througha point x in Ă M are connected by a continuous path of center leaves through x .Since the stable directions on these centers ´ verified at x vary continuously, theyall define the same direction at x . Therefore the stable funnel direction dependsonly on the point.Let V be the universal circle of the center unstable foliation W cu . For each U leaf of Ą W cu , let τ U : V Ñ S p U q be the canonical identification. Lemma 9.1.
The stable and unstable funnel directions disagree everywhere.Proof.
Since the stable and unstable funnel directions vary continuously theyeither coincide everywhere or disagree everywhere. Let us assume they coincideeverywhere. Let J be the leaf space of Ą W cu .Define a map η : J Ñ V defined as follows. Given U in Ą W cu , let q U P S p U q bethe unstable funnel point of U . Let η p U q “ p τ U q ´ p q U q .Let L be a leaf of Ą W cs and e , e distinct centers in L . Let I be the interval of J of Ą W cu leaves intersecting L in a center between e , e including the boundaryleaves. Let U, U be leaves in I intersecting L in centers c, c . Rays of c, c in thestable funnel direction are a bounded distance from each other in L , hence in Ă M . By the definition of the universal circle of the center unstable foliation, theserays define the same point in V . By hypothesis in this proof the stable funneldirection is also the unstable funnel direction in the center leaves. This impliesthat η is constant in I .By Proposition 8.3 for every U leaf of Ą W cu there is a deck translate γ p U q contained in the interior of I . Hence the union of deck translates of I is all ofthe leaf space J . This shows that η is constant. But then η would be π p M q invariant. This contradicts [FP , Proposition 5.2]. This finishes the proof. (cid:3) Lemma 9.2.
Let f be a partially hyperbolic diffeomorphism satisfying the hy-pothesis of Theorem 8.1. Let L be a leaf of Ą W cs . Then any two centers c, c in L are asymptotic in L in the stable funnel direction. In addition if two distinctcenter leaves c, c in L intersect in a point x , the following happens: if c , c arethe rays of c, c respectively starting in x and in the stable funnel direction then c “ c .Proof. Suppose that the first statement is not true. Then there are
L, c, c , and ε ą x n (say in c ) converging to the funnel point of L so that d L p x n , c q ą ε . Up to subsequence there are γ n in π p M q so that γ n p x n q converges to x . Then up to subsequence γ n p L q converges to E , γ n p c q converges toa center e in E , and γ n p c q converges to a center e in E . Since d L p p n , c q ą ε then e, e are distinct centers in E . By construction and the uniform quasigeodesicproperty, the centers e, e have the same pair of ideal points in S p E q . Thiscontradicts Theorem 8.1 that the center foliation in E is a quasigeodesic fan.This proves the second statement.To prove the second statement, suppose that there are c, c center leaves in someleaf L which intersect in x but so that the rays c , c in the stable funnel directionin L are not the same. We already know that the rays c , c are asymptotic in L . Let V be a component of L ´ p c Y c q which is between c and c . Then itcontains a stable segment s through a point y in V . As usual let ˆ f be a lift of f .Take deck translates γ i of a subsequence ˆ f n i p y q converging to y with n i Ñ ´8 ,so the stable lengths increase. Up to another subsequence suppose that γ i ˆ f n i p c i q converges to curves d , d which are contained in center leaves e , e in the limit ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 47 center stable leaf E . Let W be the limit of γ i ˆ f n i p V q which is a region between d , d . The limit of γ i ˆ f n i p s q is at least the full stable leaf s through y which iscontained in W .It could be that d , d have an endpoint, which then would be the limit z of γ i ˆ f n i p x q . In this case d , d are rays in e , e . Otherwise d , d are the full leaves e , e . In the first case the two rays of s limit to the same point in S p E q whichis the common ideal point of d , d in S p E q . But the two rays of s have to beat least some distance apart from each other or else they would intersect thesame foliated box of the center foliation, a contradiction. The rays are in theregion between e and e . This would imply that e , e are not asymptotic inthe stable funnel direction. This contradicts the first statement that has alreadybeen proved. In the second case d , d are the full leaves e , e . But then the twodistinct center leaves e , e in E have both endpoints which are the same. Thisis impossible because the center foliation is a quasigeodesic funnel in E . Thisfinishes the proof of the second statement. (cid:3) Uniqueness of the branching foliations
In this section we show:
Theorem 10.1.
Let f : M Ñ M be a partially hyperbolic diffeomorphism pre-serving branching foliations W cs and W cu such that both p f, W cs q and p f, W cu q have full pseudo Anosov behavior (Definition 6.7). Then, if W cs is anotherbranching foliation such that p f, W cs q has full pseudo-Anosov behavior then W cs “ W cs . Some parts will require less assumptions, but whenever shorter we will chooseto give a direct proof in our specific setting. Of course there is a symmetric state-ment to show uniqueness of W cu . One should compare this result to [BFFP , § W c (by intersectionbetween W cs and W cu ) and W c (by intersection between W cs and W cu ) coincide.10.1. Limit behavior.
In this section f : M Ñ M will be a partially hyperbolicdiffeomorphism preserving a branching foliation W cu tangent to E cu so that W cu is subfoliated by two f -invariant one dimensional branching foliations W c and W c tangent to E c which are quasigeodesic fan foliations (cf. Definition 8.2) obtainedby intersecting with f -invariant branching foliations W cs and W cs . In the proofof Theorem 10.1, W cs “ W cs .Notice that a priori we have four choices for funnel directions on center leaves:two unstable funnel directions (the pairs W cu , W cs and W cu , W cs ) and likewisetwo unstable funnel directions (for the same pairs). Lemma 9.1 shows that forthe same pair, the stable and unstable funnel directions are opposite.Here we show that a particular configuration holds if the foliations W c , W c donot coincide. In the next section we will show that this is impossible.For the next few results we only consider unstable funnel directions or points.So for simplicity, unless otherwise stated we refer to them as funnel directionsor funnel points. In addition the universal circle, still denoted by S univ , is theuniversal circle of W cu . Similarly, we use the previous notation for the mapsΘ L : S univ Ñ S p L q for L P Ą W cu .We need to show the following: Lemma 10.2.
The funnel points of W c and W c coincide.Proof. By Corollary 7.4 we have that the set of leaves L P Ą W cu where the funnelpoints of Ă W c and Ă W c coincide is closed and π p M q invariant. Therefore, byminimality (Proposition 8.3) we just need to show that there exist some leafwhere they coincide.To do this, take a leaf L where the funnel points differ (if there is no such leaf,there is nothing to prove). Denote by p i to the funnel point of Ă W ci in L (with i “ ,
2) and consider a point ξ P S univ with Θ L p ξ q not in t p , p u .Choose a sequence x n P L such that x n Ñ Θ L p ξ q in L Y S p L q . Now, composingwith deck transformations γ n P π p M q sending x n to a given bounded set andup to extracting a subsequence we have that γ n x n Ñ x P Ă M . Let L P Ą W cu be the limit of the leaves γ n L which is a leaf through x . The funnel points of γ n L are given by γ n p and γ n p . These converge to the funnel points in L .Since the visual measure from x n of the interval between p and p in S p L q thatdoes not contain Θ L p ξ q goes to zero with n we deduce that the endpoint of thequasigeodesic fans in L must coincide. This completes the proof. (cid:3) Finally we show the following result which is important to get a contradictionin our case, but we note that the proof may work in more generality.
Lemma 10.3.
Assume that p f, W cu q has full pseudo-Anosov behavior. If W c ‰ W c then there exists a regular pA pair p ˆ f , γ q and a leaf L P Ą W cu which is fixedby a conjugate P of P “ γ m ˝ ˆ f k (cf. Notation 4.6) such that it contains twodisjoint curves c P Ă W c and c P Ă W c whose endpoints in S p L q are Θ L images ofsuper-attracting points of P in S univ . In addition P p c i q “ c i .Proof. If W c ‰ W c , using Lemma 10.2 we know that there is a leaf E P r F such that there are center curves e P W c and e P W c in E which share bothendpoints and so that e ‰ e . Denote by p, q P S p E q the ideal points of thecurves e i ( i “ , Z between e and e as theunion of connected components of E zp e Y e q whose closure in ˆ E “ E Y S p E q is contained in E Y t p, q u . This is an open and non-empty set and we can thenconsider an unstable interval I Ă E (i.e. tangent to E u ) which is contained in Z .Fix x in the interior of I . Consider a lift f of f to Ă M . Up to subsequencethere are γ j in π p M q so that γ j f n j p x q converges to y in a leaf E , where E isthe limit of γ j f n j p E q . We can assume that γ j f n j p e i q , i “ , e i , i “ , E . This is because the curves γ j f n j p e q , γ j f n j p e q have the same pair of ideal points in S p γ j f n j p E qq and hence are a boundedHausdorff distance from each other in γ j f n j p E q . Finally γ j f n j p x q is betweenthem in γ j f n j p E q . The limit of γ n j p I q contains the full unstable leaf u of q ,which is then betweeen e , e in E .If the ideal points of u are distinct in S p E q , let u “ u , e i “ e i , E “ E .If the ideal points of u are the same point z in S p E q consider y n in u converging to z in E Y S p E q so that π p y n q converges in M . There are β n in π p M q so that β n p y n q converges to q , β n p E q converges, and we let the limitof β n p E q be E . We can also assume that β n p e i q , i “ , e i . Then β n p u q converges to at least one unstable leaf u in E which separates e from e in E . So in any case we obtain a leaf E of Ą W cu with ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 49 two centers e , e of Ă W ci respectively so that e , e have the same ideal points in S p E q and there is an unstable leaf u in L separating e from e .We assumed that p f, W cu q has full pseudo Anosov behavior (cf. Definition6.7). So there is a conjugate P of the full regular pair so that points in disjointrays of e i are either in distinct basins of attraction of P or distinct basins ofrepulsion of P . In the case of attraction (repulsion) use iterates P n as n Ñ 8 (as n Ñ ´8 ). In either case we get L is the limit of P n p E q , and up to subsequence P n p e i q converges to c i center leaves in L which are invariant by P and there is anunstable leaf u in E separating c from c and u invariant by P . Since P p u q “ u the ideal points can only be super attracting.This finishes the proof of the Lemma. (cid:3) Orientability of the center foliations.
Even if the funnel points coin-cide (cf. Lemma 10.2), the orientation may be different in both. This will play acrucial role in the proof, so we introduce the following definition:
Definition 10.4.
Let T be a quasigeodesic fan foliation of F (cf. Definition 8.2)and consider an orientation of the tangent space to T . We say that T is oriented towards the funnel point if every curve of r T X L is oriented in the direction of thefunnel point. Otherwise, we say that T is oriented against the funnel direction. Remark . Notice that the definition makes sense. First of all, the fact that T is a quasigeodesic fan foliation implies that T is orientable. Secondly, the funnelpoint varies continuously (cf. Corollary 7.4), therefore, either r T X L is orientedin the direction of the funnel point everywhere or nowhere.We will choose from now on an orientation in E c making that W c is orientedtowards the funnel point. Lemma 10.6.
Assume that p f, W cu q has full pseudo-Anosov behavior. If W c ‰ W c then W c is oriented against the funnel point.Proof. We work in the leaf L given by Lemma 10.3 where we have disjoint centers c i P Ă W ci fixed by P which join the attracting points a , a of P in S univ and whichare separated by a fixed unstable leaf u which also joins those points. Let x P u be the unique fixed point. Note that since both W c and W c share their funnelpoints, we can assume that a is the funnel point for both.Let e i be a curve of Ă W ci through the point x P u and fixed by P . Note that x may belong to many curves in Ă W ci but at least one must be fixed by P , we chooseany such fixed curve. It is important to remark that e i ‰ c i since c i does notintersect u .Consider the ray of e i from x pointing to the region between u and c i . Thisregion has limit points only a and a . We claim that the endpoint of the raymust be a : if it were a this is different than a and so the other endpoint of e i would be a . Hence e i is another curve in Ă W ci from a to a (besides c i ) and thisis inconsistent with being a quasigeodesic fan foliation.Since the regions between u and c and u and c are oriented differently from x , we deduce that the orientation of e has to be against the funnel point. Sinceorientations coincide or disagree everywhere, this concludes. (cid:3) Proof of Theorem 10.1.
We first show:
Lemma 10.7. If W cs “ W cs ‰ W cs then there is a leaf U P Ą W cu such that thefoliations Ă W c and Ă W c are different. xua a r r c e e c Figure 8.
Proof of Lemma 10.6.
Proof.
We show that if Ă W c “ Ă W c in every leaf U of Ą W cu , then W cs “ W cs . Let L in W cs and U in W cu intersecting L in a center c leaf of Ă W c . Since Ă W c “ Ă W c ,then there is E leaf of W cs intersecting U also in c . We will show that E “ L ,hence every leaf of W cs is also a leaf of W cs and vice versa, proving the result.Let W be the union of the stable leaves intersecting c . The foliations W cs , W cs have leaves which are stable saturated, hence W is contained in both L and E .Let p be the (stable) funnel point of L . Lemma 9.2 shows that for any othercenter c in L then c, c are asymptotic in the direction of p . Hence c has a raytowards p contained in W (so contained in E ). This ray defines direction 1 in c .Let V be a leaf of Ą W cu so that V X L “ c . Let c “ V X E . Then c , c sharea ray in direction 1. The unstable funnel direction in V induces the oppositedirection (direction 2) in c by Lemma 9.1. It follows that in V the rays of c , c corresponding to direction 2 have the same ideal point q in S p V q .Notice that c is a leaf of Ă W c and c is a leaf of Ă W c . Since these foliations arethe same in V , then c is also a leaf of Ă W c in V . But then c , c are leaves of Ă W c with same pair of ideal points in V (in direction 1 they share a ray, in direction 2they both limit to q ). Since Ă W c is a quasigeodesic fan in V it follows that c “ c .In other words c is contained in E . Since this is true for any center in L then L Ă E . Since L is properly embedded this implies that L “ E . This finishes theproof. (cid:3) Now we can apply what we showed before to prove uniqueness:
Proof of Theorem 10.1.
By the previous Lemma it is enough to show that W c “ W c , so assume by way of contradiction that W c ‰ W c . Now use Lemma 10.3,which provides a P and a leaf L of Ą W cu fixed by P , containing two leaves c i of W ci invariant by P and an unstable leaf u in L fixed by P separating c from c in L . This is the setup of Lemma 10.6 and we use the same center curves e i through a fixed point x P u as in that lemma.Recall the setup of Lemma 10.6: there are 4 fixed points of P on S p L q , whichare a , a (attracting) and r , r (repelling). Since e is fixed by P its ideal pointsare fixed points of P in S p L q . One of them is a . The other ideal point z of e a a r yc I J Vc I c e Figure 9.
Proof of uniqueness. cannot be r as c separates r from e . The point z cannot be a either since c already has ideal point r . It follows that z “ r , see figure 9. Here r is therepelling fixed point of P acting on L Y S p L q which is not separated by u from c . In particular, e must intersect c .Let y P c X e be the last point of intersection (when following e towards thepoint r or c towards a which are the positive orientations). The point y mustbe fixed by P . Let I be the ray of c from y to the ideal point a , and let I be the ray of e from y to the ideal point r . It follows that I Y I separates L in two components, one of which, that we call Z has its closure in ˆ L containingthe segment in S p L q from r to a (and not intersecting any other of the fixedpoints of P in S p L q ). Notice that I is in a leaf c of Ă W c and at y the orientationin c is pointing away from I . Conversely I is contained in a leaf E of Ă W c andat y its orientation is pointing into I ´ in other words pointing away from I .This is because Lemma 10.6. Since both I and I are oriented coherently at y it follows that the unstable manifold u p y q of y has one ray J inside Z . But J isinvariant under P and it is expanding under P , hence J must have ideal point a . Consider the region V of L bounded by the union of I and the ray J .If c P Ă W c is a curve intersecting V it follows that it must intersect u p y q twice,contradicting the fact that an unstable manifold cannot intersect the same leafof Ą W cs twice (cf. § (cid:3) Hyperbolic manifolds: Proof of Theorems A and B
In this section f : M Ñ M will be a partially hyperbolic diffeomorphism ofa hyperbolic 3-manifold. Recall that a hyperbolic manifold is one obtained asa compact quotient of (cid:72) by isometries. By Perelman’s proof of Thurston’s ge-ometrization conjecture this is equivalent to being aespherical (i.e. π p M q “ t u )and homotopically atoroidal (i.e. no π -injective torus) with infinite fundamen-tal group (see [BFFP , Appendix A]). Note that we will only use the atoroidalcondition plus generalities about foliations. Dichotomy: Discretized Anosov or double translation.
Here weexplain how the main results of [BFFP ] allow us to reduce the proof of TheoremsB to what we did so far.Up to finite cover and iterate, we have that f must preserve branching foliationsand be homotopic to the identity (this is because of Mostow rigidity, see e.g.[BFFP , Proposition A.3]). We will lift this assumptions in § ] we will use. See [BFFP , Theorem 2.4]. Theorem 11.1.
Let f : M Ñ M be a partially hyperbolic diffeomorphism ho-motopic to the identity of a hyperbolic 3-manifold M which preserves branchingfoliations W cs and W cu . Then, (i) either f is a discretized Anosov flow, or, (ii) the pairs p f, W cs q and p f, W cu q have the commuting property (cf. § Notice that in case (ii) of this theorem we are in option (ii) of § r f of f to Ă M which commutes with all deck transformations and r f acts freely on the leaf space of Ą W cs .We need to make some comments to explain how this follows directly from[BFFP ]. When f is homotopic to the identity we call a lift r f : Ă M Ñ Ă M a goodlift if it commutes with all deck transformations. Such a lift can be obtained bylifting a homotopy to the identity. See [BFFP , Definition 2.3]. The good lifthas the property required for p f, W cs q , p f, W cu q to have the commuting property(the fact that W cs and W cu are (cid:82) -covered and uniform are direct consequencesof [BFFP , Theorem 2.4]).Notice that if f is a discretized Anosov flow then it is a collapsed Anosov flow,so we need to analyse only the second situation which we call double translation .11.2. Regulating pseudo-Anosov flows.
We state here the results that followfrom [Th , Ca , Fen ]. We remark that these results depend only on the fact that M is atoroidal and not on its geometry (said otherwise, they depend on the coarsegeometry and not on the precise hyperbolic metric).See [BFFP , Proposition 10.1] for the adaptation to branching foliations: Theorem 11.2.
Let F be a (cid:82) -covered, transversely oriented and uniform branch-ing foliation of a hyperbolic 3-manifold M . Then, there exists a pseudo-Anosovflow Φ t : M Ñ M transverse and regulating to F . Recall that the condition of being regulating means that in the universal cover Ă M , given two leaves L, L P r F there is a uniform time t : “ t p L, L q such thatfor every x P L it holds that Φ t p x q P L for some | t | ă t . We state the followingrelevant properties about pseudo-Anosov flows in hyperbolic 3-manifolds thatfollow from previous work by several authors (we give a very short sketch of theproof pointing to some references for more details): Theorem 11.3.
Let Φ t : M Ñ M be a pseudo-Anosov flow in a hyperbolic 3-manifold. Then, Φ t is transitive and therefore both the weak stable and weakunstable (singular) foliations are minimal. Moreover, if Φ t is regulating to an (cid:82) -covered foliation it cannot be an Anosov flow.Proof. If a pseudo-Anosov flow in a 3-manifold is not transitive, then it has anincompressible torus transverse to the flow (see [Mos]). Since M is hyperbolicthen this is impossible.Once that a pseudo-Anosov flow is transitive, the minimality of the singularfoliations follows easily by the Anosov closing lemma and the inclination lemma. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 53
Finally, in [BFFP , Proposition D.4] we explain how the fact that pseudo-Anosovflows transverse and regulating to (cid:82) -covered foliations in hyperbolic 3-manifoldscannot be Anosov follows from previous work by Barbot and the first author. (cid:3) We note that this provides pA-pairs for p f, W cs q and p f, W cu q in case (ii)of Theorem 11.1 since for every periodic orbit of the transverse and regulatingpseudo-Anosov flow one can construct a pA-pair associated to it using the decktransformation associated to the orbit. Note that r f acts trivially in the universalcircle, so one needs only to care about the action of the deck transformation (see[BFFP , Proposition 10.2]). Using Corollary 5.3 one deduces that both p f, W cs q and p f, W cu q have the periodic commuting property. In the next lemma we showthat this produces pA pairs. Remark . Let γ be a deck transformation. A point p P S univ is superattractingfor γ (the induced action on S univ ), if and only if for some (and hence for any) L in r F if γ L is an expression in L of the action of γ , then the following happens:there is a neighborhood basis of Θ p p q in L Y S p L q defined by geodesics (cid:96) i in L sothat the minimum distance in L between points in (cid:96) i and γ L p (cid:96) i q goes to infinitywith i . See also proof of Proposition 3.7. Lemma 11.5.
Let Φ t be a pseudo-Anosov flow transverse to F as in Theorem11.3. Let γ be a deck transformation associated with a periodic orbit of Φ t . Thensome power of γ has fixed points in S univ and so that all fixed points are eithersuper attracting or super repelling.Proof. We follow the setup in [BFFP ]. Fix L in r F . Let G sL , G uL be the singularone dimensional foliations in L induced by intersecting the stable and unstable2-dimensional singular foliations of Φ t lifted to Ă M with L . The non singularleaves are uniform quasigeodesics [BFFP , Fact 8.3]. Let B s , B u be the geodesiclaminations in L obtained by pulling tight the leaves of G sL , G uL respectively. Eachnon singular leaf of G sL is a uniformly bounded Hausdorff distance in L from aunique leaf of B s . A p -prong leaf of G sL generates p leaves of B s .The deck transformation γ is associated with a periodic orbit α of Φ t and fixesa lift r α to Ă M . Up to taking a power assume that γ fixes all prongs of r α . Assumethat γ is associated with the negative direction of α . As in [BFFP , Section 8]let τ : L Ñ γ ´ p L q be the map obtained by flowing x in L along its r Φ t flow lineuntil it hits γ ´ p L q . Notice that d Ă M p x, τ , p x qq , x P L is bounded. Then γ ˝ τ isa representative in L of the action of γ . Let h “ γ ˝ τ .Let x “ r α X L , which is the only fixed point of h “ γ ˝ τ . Fix an unstableprong η of x with ideal point p in S p L q . We will prove that p is a super attractingfixed point of γ . For a stable prong we get a super repelling fixed point. Up toapplying a power of γ we can assume that the Hausdorff distance between L and γ p L q is very big. This is okay since the lemma claims the result for a power of γ . Then by [BFFP , Fact 8.4] the map h expands length along G uL exponentiallyand contracts length along G sL exponentially (see also [Fen ]). This means thatlength along η from y to h p y q goes to infinity as y escapes in η . We consider abasis neighborhood of p defined by leaves of G sL intersecting η : given y in η let (cid:96) y the leaf of G sL through y .Given y in η let g y be the geodesic associated with (cid:96) y : it is a bounded Haus-dorff distance in L from (cid:96) y . Let ν be the geodesic in L associated with G uL p x q (forsimplicity assume x is non singular, otherwise there are 2 such geodesics associ-ated with the ray η ). Then the angle between ν and any g y is bounded below by a ą
0. Also the y is a bounded distance from the intersection between ν and g y . These facts imply that the minimum distance between points in (cid:96) y and h p (cid:96) y q goes to infinity as y escapes in η .This proves that p is a superattracting point. This finishes the proof. (cid:3) Remark . Note that the pseudo-Anosov flows associated to W cs and W cu given by Theorem 11.2 may be different and not even share the same homotopyclasses of periodic orbits. This will not be an issue, and we will obtain a posteriori,that both pseudo-Anosov flows are orbit equivalent since this is the case alwaysfor the weak stable and unstable foliations of an (cid:82) -covered Anosov flow in ahyperbolic 3-manifold.11.3. Existence of full pseudo-Anosov pairs.
Here we show:
Proposition 11.7.
Let f : M Ñ M be a partially hyperbolic diffeomorphism ofa hyperbolic 3-manifold with f homotopic to the identity and preserving trans-versely oriented branching foliations W cs and W cu . Suppose that both p f, W cs q and p f, W cu q have the periodic commuting property. Then, both pairs have fullpseudo-Anosov behavior (cf. Definition 6.7). In particular, f is a collapsedAnosov flow.Proof. This follows from the existence of a regulating pseudo-Anosov flow. Wediscuss the arguments to get the statements in our current framework. The factthat p f, W cs q and p f, W cu q have the periodic commuting property follows fromCorollary 5.3 and [BFFP , Proposition 10.2] as explained in the previous section.Let Φ cst be the pseudo-Anosov flow given by Theorem 11.2 for the branchingfoliation W cs (the same arguments apply for W cu ). To obtain the existence ofa full pA pair (cf. Definition 6.5) we use the fact that the singular foliations ofthe pseudo-Anosov flow are minimal. The good pairs we will be using are p r f , γ q where γ is a deck transformation associated with a regular periodic orbit of Φ cst and r f is the good lift of f to Ă M . Since Φ cst is regulating for W cs , then γ actsfreely on the leaf space of Ą W cs . Hence p r f , γ q is a good pair. Up to a power assumethat γ preserves all the prongs of the periodic orbit when lifted to the universalcover. Lemma 11.5 implies that any P “ r f m γ n ( n non zero) has periodic pointswhen acting on the universal circle of W cs . If there are fixed points then they areall either super attracting or super repelling if | n | is sufficiently big. Hence p r f , γ q is a regular pA-pair for p f, W cs q .Now we explain why this provides a full pair. For each leaf L of Ą W cs let B sL , B uL be the geodesic laminations in L obtained by pulling tight in L the leaves of thestable and unstable foliations of r Φ cst intersected with L . The complementaryregions of each of these geodesic laminations in L are finite sided ideal polygons,and the complementary regions of the union are relatively compact polygons withbounded diameter. The union of these over L projects to transverse laminationsin M ´ for details on these laminations see [Fen ] . These laminations areminimal. For each ε ą d ą d in any of these laminations are ε dense in M . Choose ε much smaller thanthe product foliation size of all the foliations or laminations involved. Given thedeck transformation γ associated to a regular periodic orbit µ , then the stableand unstable leaves of µ are annuli or M¨obius bands producing like sets in the In [Fen ] the leafwise geodesic laminations are constructed first, before the pseudo-Anosovflow, via an analysis of the action of π p M q on the universal circle of the foliation W cs . Thenthese laminations blow down to singular foliations producing a pseudo-Anosov flow. In [Fen ]this is worked out for (non branching) foliations. The case of W cs a branching foliation is workedout in [BFFP ]. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 55 leafwise geodesic laminations. A fixed annulus or M¨obius (denoted by A s , A u )band near the blow up of the periodic orbit is ε dense in M . The A s , A u intersectin a core closed curve corresponding to the blow up (or pre-image) of the periodicorbit µ .Let now η be a geodesic ray in L . By the above there is a length d ą ě d in η intersects one of the laminations B sL or B uL .There is α ą B sL or B uL has angle ą α . This implies that η intersects either a lift of A s or A u making an angle ą α . This lift is given by a deck translate β ´ of a fixed lift of either A s or A u .This implies that the conditions of Definition 6.5 are satisfied.After we showed that both pairs have full pseudo-Anosov behavior, the factthat f is a collapsed Anosov flow follows from Theorem 8.1. (cid:3) Proof of Theorems A and B.
Theorem A follows immediately fromTheorem B since the existence of a collapsed Anosov flow in M explicitely asksfor the existence of a (topological) Anosov flow in M . Notice that in hyperbolicmanifolds every topological Anosov flow is transitive, and therefore the existenceof a topological Anosov flow implies the existence of an Anosov flow (cf. § Proof of Theorem B.
As explained we can assume that if f : M Ñ M is a par-tially hyperbolic diffeomorphism in a hyperbolic 3-manifold, then Theorem Bholds for the lift of some iterate of f to a finite cover (see Theorem 11.1 andProposition 11.7). We denote the finite cover of M as M and f to the lift of thefinite iterate of f to M . We emphasize that the finite cover is considered so thatall bundles are orientable, in the double translation case we will show a posteriorithat this finite cover is indeed not necessary as the bundles were orientable inthe first place. Up to taking a further cover and lift of further of iterate we mayassume that M is a regular cover of M .We want to show that f preserves branching foliations so that Theorem 11.1applies and this completes the proof together with Proposition 11.7.For this, we lift the branching foliations W cs , W cu preserved by f to Ă M whichis the common universal cover of M and M and denote the lifts as Ą W cs and Ą W cu .Let r f the good lift of f to Ă M . We need to show first that deck transformations π p M q preserve Ą W cs and Ą W cu (we know that the subgroup π p M q ă π p M q doespreserve them).We first assume that we are in the situation of Theorem 11.1 (ii).We consider then the pair of foliations W cs and W cu in M obtained by pro-jecting to M the foliations γ Ą W cs and γ Ą W cu for some γ P π p M q . The reasonwhy these project to M is because π p M q is a normal subgroup of π p M q so π p M q preserves γ Ą W cs , γ Ą W cu . Since r f commutes with all deck transformations,then f preserves W cs , W cu . By Theorem 10.1 it is enough to show that the pairs p f , W cs q and p f , W cu q have full pseudo-Anosov behavior. But this follows as inProposition 11.7 once we show that r f acts as a translation on γ Ą W cs and γ Ą W cu which is direct since r f commutes with γ .Since the foliations are invariant by deck transformations of M , and r f actsas a translation and commutes with deck transformations, it follows that decktransformations of π p M q must preserve the orientation transverse to both Ą W cs and Ą W cu . Since the center direction is orientable because of the existence of a funnel point (that must also be preserved by deck transformations) we deducethat all bundles were orientable in M and therefore the finite lift was not necessaryto make the bundles orientable.Finally, in this case, taking the iterate is not necessary. For this it is enough toshow that the foliations f p W cs q and f p W cu q are equal to W cs and W cu , but thisfollows by the same argument applying Theorem 10.1. (See also [BFP, TheoremB].)This finishes the analysis of the case when r f acts as a translation in the leafspaces of Ą W cs , Ą W cu .We now deal with the case that r f fixes every leaf of Ą W cs and of Ą W cu . Here weuse [BFFP , Theorem 12.1]. It shows that f is dynamically coherent preservingactual foliations, center stable and center unstable. The center foliation is theintersection of these, and hence it is preserved by f as well. In addition [BFFP ,Theorem 12.1] shows that a finite iterate of f is a discretized Anosov flow pre-serving each leaf of the center foliation. In this case let h be the identity. Theself orbit equivalence β is f itself since it preserves the center foliation. Orbits ofthe flow are tangent to the center direction, showing that f is a collapsed Anosovflow.This completes the proof of Theorem B. (cid:3) Unique integrability properties.
We state here a strong geometric con-sequence of our study:
Theorem 11.8.
Let f : M Ñ M be a partially hyperbolic diffeomorphism in ahyperbolic 3-manifold. Then, f admits a unique pair W cs , W cu of f -invariantbranching foliations tangent respectively to E cs and E cu . Moreover, every curve c tangent to E c in Ă M is contained in the intersection of a leaf L P Ą W cs and a leaf F P Ą W cu (which is connected).Proof. Suppose that f k is a positive iterate homotopic to the identity and let g “ f k . Let r g be the good lift to Ă M . We start by proving uniqueness of thebranching foliations.Suppose first g, W cs , W cu is a double translation and suppose that f preservesanother branching foliation W cs . Then g also preserves W cs . Mixed behavior ingeneral means that r g fixes leaves of one foliation (of the pair Ą W cu , Ą W cs ), but noneof the other. But mixed behavior is impossible in hyperbolic 3-manifolds [BFFP , § r g acts as a translation on Ą W cu , then r g also acts as a translation on Ą W cs so W cs , W cu is a double translation pair. Then p g, W cs q , p g, W cu q have the periodiccommuting property (cf Def. 4.5). Theorem 10.1 implies that W cs “ W cs .Suppose now that g, W cs , W cu is a discretized Anosov flow and let W cs pre-served by f . Then g also preserves W cs . Again mixed behavior cannot occur,and now r g fixes every leaf of Ą W cu , so it fixes every leaf of Ą W cs . It follows that g, W cs , W cu is also a discretized Anosov flow. Then W cs “ W cs follows from[BFFP , Lemma 7.6].The statement about curves tangent to E c is proved from uniqueness of branch-ing foliations [BFP, Proposition 10.6] as follows: Let c be a curve tangent to E c .Following previous notation let f be a lift of a a finite iterate of f to a finitelift M of M so that all bundles are orientable in M and f preserves the ori-entability of the bundles. In addition suppose the original finite iterate of f ishomotopic to the identity. Then c lifts to c in M tangent to the center bundle.[BFP, Proposition 10.6] requires the orientability of the bundles which is attained ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 57 by f in M , hence c is obtained as the intersection of a leaf of the center stablefoliation and a leaf of the center unstable foliation in M . But we proved thatthese foliations in M project to W cs , W cu in M . This proves the result for curvestangent to E c . (cid:3) Immediate consequences are the following:
Corollary 11.9.
Let f : M Ñ M be a partially hyperbolic diffeomorphism ina hyperbolic 3-manifold. Then f is a discretized Anosov flow if and only if thebundle E c is (uniquely) integrable. This follows because in [BFFP , Theorem B] we prove that for double trans-lations E c cannot integrate to a foliation. By the uniqueness properties given byTheorem 11.8 the result follows.One can also get a result in the direction of the plaque expansivity conjecture[HPS] in a concrete setting. Corollary 11.10.
Let f : M Ñ M be a diffeomorphism of a hyperbolic 3-manifold so that T is a one-dimensional normally hyperbolic foliation preservedby f . Then f is dynamically coherent and plaque expansive. We refer the reader to [HPS] for a definition of T being a one-dimensionalnormally hyperbolic foliation, which in particular implies that f is partially hy-perbolic. Proof.
Theorem 11.8 shows that f preserves a unique pair of branching foliations W cs , W cu and any curve tangent to E c is contained in the intersection of a leaf of W cs and a leaf of W cu . It follows that T has to be the center foliation associatedwith these branching foliations. Since T is a foliation (as opposed to a branchingone dimensional foliation) it follows that W cs , W cu are also foliations, and do nothave branching. This shows that f is dynamically coherent.Using Theorem B we get that an iterate of f is a discretized Anosov flow.These are plaque expansive [Mart]. (cid:3) Seifert manifolds: Proof of Theorem C
In this section we consider a partially hyperbolic diffeomorphism f : M Ñ M where M is a Seifert manifold and such that the induced action of f in the base ispseudo-Anosov. We note here that in contrast with the hyperbolic case (TheoremB) the arguments here do not rely on [BFFP ] and this result can be consideredself contained.12.1. Existence of full pseudo-Anosov pairs.
We first consider a finite cover M of M which makes all bundles orientable and take an iterate of f which liftsto the finite cover and preserves the orientation of the bundles. We call thisdiffeomorphism f : M Ñ M . The fact that M is Seifert and the action inthe base is pseudo-Anosov is unchanged, and we can apply Theorem 2.1 to getbranching foliations W cs and W cu invariant under f . We may assume withoutloss of generality that M is a circle bundle over a surface by taking a furtherfinite cover. We denote by π : M Ñ M the finite cover we chose.Using [HaPS, § horizontal , in par-ticular, they are (cid:82) -covered and uniform and by hyperbolic leaves. Moreover, itfollows that in Ă M , the universal cover of M the action of δ P π p M q associatedto the fiber of the circle bundle acts freely on the leaf space of both Ą W cs and Ą W cu .Using Thurston’s classification of surface diffeomorphisms [Th ] one deduces: Proposition 12.1.
The pairs p f , W cs q and p f , W cu q have full pseudo-Anosovbehavior.Proof. This follows from [CB, Lemmas 6.2 and 6.4] the same way as in Proposition11.7. (cid:3)
We deduce from Theorem 8.1:
Corollary 12.2.
The diffeomorphism f is a collapsed Anosov flow. Proof of Theorem C.
This follows exactly as Theorem B, one showsthat after the finite cover and iterate to preserve branching foliations, these areinvariant under the deck transformations using Theorem 10.1 (and so descend to M ) and then that taking an iterate is also not necessary. Remark . One also obtains unique integrability results analogous to those ofTheorem 11.8. 13.
Further results
In this section we give a couple of applications of pseudo-Anosov pairs topartially hyperbolic diffeomorphisms in other 3-manifolds or isotopy classes toshow the flexibility of the tool. We hope other applications can be found.13.1.
General partially hyperbolic diffeomorphisms homotopic to theidentity.
Theorem 11.2 in [Ca , Fen ] for atoroidal manifolds has been extendedrecently by the first author to more general manifolds [Fen ]. In particular, it willallow us to extract the following result that holds in a larger class of 3-manifolds: Theorem 13.1.
Let F be a transversely oriented, (cid:82) -covered, uniform foliationon a 3-manifold with an atoroidal piece. Then, there exists a deck transformation γ P π p M q which acts as a translation on the leaf space of r F and the inducedaction in the universal circle S univ of F has exactly exactly fixed points: twosuper attracting and two super repelling fixed points. As a consequence, we get:
Theorem 13.2.
Let f : M Ñ M is a partially hyperbolic diffeomorphism ho-motopic to the identity on a 3-manifold having some atoroidal piece in the JSJdecomposition preserving a branching foliation W cs so that the good lift r f of f isa translation on the leaf space of Ą W cs . Then both the center (branching) foliationand the strong stable foliation have small visual measure inside the leaves of W cs (cf. Theorem 5.6). In particular for any ray r of a center leaf c in a leaf L of Ą W cs , then r accumulates in a single point in S p L q .Proof. By translation we mean it has no fixed points on the leaf space of Ą W cs .This was analyzed in Proposition 4.6 of [BFFP ], where it is proved that thisimplies that W cs is (cid:82) -covered and uniform. The translation of r f also impliesthat W cs is transversely orientable. We can apply Theorem 13.1 and we get that p f, W cs q has the periodic commuting property and it has at least one (regular)pA pair. Therefore, Theorem 5.6 applies and we get the statement. (cid:3) We note that for discretized Anosov flows the center foliation also has smallvisual measure, but the strong stable does not, which looks as something quiteremarkable about this result that needs to be better understood. The previousresult complements well with [BFFP , Theorem 1.2]. ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 59
We now explain the proof of Theorem 13.1. This is proved in Proposition5.2 of [Fen ]. We give an alternate proof of super attracting/repelling behaviorwhich uses less of the transverse regulating flow and the transverse laminationand relies only on large scale geometry. We put this alternate proof here as it maybe useful in other contexts. In particular the proofs of Claim 13.3 and Claim 13.4work even when the deck transformation γ has two fixed points in the universalcircle S univ . The super attracting property proved in Proposition 5.2 of [Fen ]only works for γ associated with an orbit of the flow which necessarily has (upto finite iterate) at least four fixed points in S univ . Proof of Theorem 13.1.
For simplicity we assume that M is orientable, whichcan be accomplished by taking a double cover. We will divide the proof in threesteps. We assume some background on 3-manifolds, see [BFFP , Appendix A]and [BFFP , Appendix A]. First we show the following claim reminiscent of[BFFP , Lemma 8.5]. Recall that for leaves L, E P r F we have a quasi-isometry τ L,E : L Ñ E given by Proposition 2.4. Claim 13.3.
Let γ P π p M q be a deck transformation of M acting increasinglyin the leaf space of r F and such that γ fixes an atoroidal piece P and does notfix the lift of a JSJ tori. Then for every R ą there is K ą such that if L P r F is some leaf and we denote g : L Ñ L to be the quasi-isometry given by γ ˝ τ L,γ ´ L : L Ñ L then there is a disk D of radius R in L such that if y R D then d p y, g p y qq ą K .Proof. Notice that γ is in π p P q and does not represent a peripheral curve in P . The proof is the same as [BFFP , Lemma 8.5] once one notices that thehypothesis on γ force the existence of an axis for the action on the atoroidalpiece (which admits a hyperbolic structure). This also follows from an argumentsimilar to Lemma 11.5 using the laminations constructed in [Fen ]. (cid:3) Now, using some hyperbolic geometry on the leaves we can show:
Claim 13.4. If γ is a deck transformation as in the previous claim, then everyfixed point of γ acting on S univ is either super attracting or super repelling.Proof. Take ξ P S univ and assume that it is fixed by the action of γ . Considera leaf L P r F and a geodesic ray r whose endpoint is ˆ ξ “ Θ L p ξ q in S p L q . Let g “ γ ˝ τ L,γ ´ p L q . It extends to a homeomorphism of L Y S p L q still denoted by g . Since the action of γ in S univ is given by the action of g as defined above in S p L q via the identification of Θ L we get that g p r q is a quasi-geodesic ray thatalso lands in ˆ ξ . Let r be the geodesic with same starting point and ideal pointas g p r q . Notice that r is asymptotic with r .Fix a sequence of neighborhoods of ˆ ξ in S p L q given by intervals r a n , b n s in S p L q so that the geodesics α n joining a n , b n converge to ˆ ξ and are orthogonal to r . It follows that g p α n q is a quasigeodesic which makes a uniform (coarse) anglewith g p r q . In other words if (cid:96) n is the geodesic in L with same ideal points as g p α n q then the angle between (cid:96) n and r is bounded below by a ą
0. This is becauseif the angle goes to 0, then one gets points x n , y n in r , (cid:96) n respectively which arevery close in L and very far away from the intersection of r , (cid:96) n . In addition x n converging to Θ L p ξ q . Pulling back by g ´ (using that r , r are asymptotic) onegets points in r , α n which are boundedly close in L but the points in α n very farfrom r . This is a contradiction to g being a quasi-isometry..Using the previous claim we obtain the desired result. See [BFFP , LemmaA.10] for a similar argument in a slightly different setting. (cid:3) Finally, [Fen , Proposition 5.2] gives a deck transformation fixing an atoroidalpiece and with at least four fixed points at infinity. This completes the proof. (cid:3) Remark . Note that in the setting of the Theorem 13.2 we also get that f cannot be dynamically coherent (see Remark 5.2).13.2. Results in Seifert manifolds with only one pseudo-Anosov compo-nent.
There is also a partial statement similar to Theorem 13.2 where we replaceTheorem 13.1 with the results in [BFFP , Appendix A]. Theorem 13.6.
Let f : M Ñ M be a partially hyperbolic diffeomorphism of aSeifert manifold so that the induced action on the base has some pseudo-Anosovcomponent preserving branching foliations W cs and W cu , which are horizontal.Then, both the center (branching) foliation and the strong stable foliation havesmall visual measure inside the leaves of W cs (cf. Theorem 5.6). We note that this result is new even for the examples of [BGHP] where this be-havior of the strong foliations was unknown. The horizontality condition impliesin particular that W cs , W cu are (cid:82) -covered, which is needed to apply the results inthis article (see [HaPS] for conditions under which the assumption is met). Notethat incoherence in this setting (cf. Remark 13.5) was shown in [BFFP ]. References [Ba ] T. Barbot, Caract´erisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles. Ergodic Theory Dynam. Systems (1995), no. 2, 247–270. (Cited on page 14.) [Ba ] T. Barbot, De l’hyperbolique au globalement hyperbolique. Habilitation `a Diriger desRecherches. Universit´e Claude Bernard - Lyon I, 2005. (Cited on pages 6 and 14.) [BaFe ] T. Barbot, S. Fenley, Pseudo-Anosov flows in toroidal manifolds. Geom. Topol. (2013), no. 4, 1877–1954. (Cited on page 6.) [BaFe ] T. Barbot, S. Fenley, Classification and rigidity of totally periodic pseudo-Anosov flowsin graph manifolds. Ergodic Theory Dynam. Systems (2015), no. 6, 1681–1722. (Citedon page 6.) [BaFe ] T. Barbot, S. Fenley, Free Seifert pieces of pseudo-Anosov flows, Geom. Topol. , toappear. (Cited on page 6.) [Bart] T. Barthelm´e, Anosov flows in 3-manifolds, lecture notes for the School on contemporarydynamical systems, Montr´eal (2017). Available at the authors web page. (Cited on pages 6and 14.) [BFFP] T. Barthelm´e, S. Fenley, S. Frankel, R. Potrie, Research Announcement: Partiallyhyperbolic diffeomorphisms homotopic to the identity on 3-manifolds, (2020) 341–357. (Cited on pages 2 and 7.) [BFFP ] T. Barthelm´e, S. Fenley, S. Frankel, R. Potrie, Partially hyperbolic diffeomor-phisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case,arXiv:1908.06227. (Cited on pages 2, 7, 14, 15, 22, 29, 44, 51, 52, 53, 57, 59, and 60.) [BFFP ] T. Barthelm´e, S. Fenley, S. Frankel, R. Potrie, Partially hyperbolic diffeomorphismshomotopic to the identity in dimension 3, Part II: Branching foliations, arXiv:2008.04871. (Cited on pages 2, 3, 4, 6, 7, 8, 9, 15, 19, 25, 29, 45, 47, 52, 53, 54, 56, 57, 58, and 59.) [BFFP ] T. Barthelm´e, S. Fenley, S. Frankel, R. Potrie, Dynamical incoherence for a largeclass of partially hyperbolic diffeomorphisms, to appear in Ergodic Theory Dynam. Systems (Cited on pages 2, 4, 5, 7, 13, 19, 29, 59, and 60.) [BFP] T. Barthelm´e, S. Fenley, R. Potrie, Collapsed Anosov flows and self orbit equivalences,arXiv:2008.06547 (Cited on pages 1, 2, 3, 4, 6, 7, 8, 9, 14, 44, 45, and 56.) [BBY] F. Beguin, C. Bonatti, B. Yu, A spectral-like decomposition for transitive Anosov flowsin dimension three,
Math. Z. (2016), no. 3-4, 889–912. (Cited on page 6.) [BBP] C. Bonatti, J. Bowden, R. Potrie, Some remarks on projectively Anosov flows in hyper-bolic 3-manifolds, (2020) 359–369. (Cited on page 6.) [BGHP] C. Bonatti, A. Gogolev, A. Hammerlindl, R. Potrie, Anomalous partially hyperbolicdiffeomorphisms III: abundance and incoherence,
Geom. Topol. (2020), no. 4, 1751–1790. (Cited on pages 1, 2, 4, 30, 32, and 60.) ARTIAL HYPERBOLICITY AND PSEUDOANOSOV DYNAMICS 61 [BGP] C. Bonatti, A. Gogolev, R. Potrie, Anomalous partially hyperbolic diffeomorphisms II:stably ergodic examples.
Invent. Math. (2016), no. 3, 801–836. (Cited on page 1.) [BPP] C. Bonatti, K. Parwani, R. Potrie, Anomalous partially hyperbolic diffeomorphisms I:dynamically coherent examples,
Ann. Sci. ´Ec. Norm. Sup´er. (4) (2016), no. 6, 1387–1402. (Cited on page 1.) [BW] C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology (2005) (2005), no. 3, 475–508. (Cited on page 1.) [BM] J. Bowden, K. Mann, C -stability of boundary actions and inequivalent Anosov flows,arXiv:1909.02324 (Cited on page 6.) [BH] M. Bridson, A. Haefliger, Metric spaces of non-positive curvature.
Grundlehren der Math-ematischen Wissenschaften, . Springer-Verlag, Berlin, 1999. xxii+643 pp. ISBN: 3-540-64324-9 (Cited on pages 9, 10, 12, and 41.) [Bru] M. Brunella, Separating the basic sets of a nontransitive Anosov flow.
Bull. London Math.Soc. (1993), no. 5, 487–490. (Cited on page 14.) [BI] D. Burago, S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelianfundamental groups. J. Mod. Dyn. (2008), no. 4, 541–580. (Cited on pages 3, 5, 6, 7, and 8.) [Ca ] D. Calegari, The geometry of R -covered foliations. Geom. Topol. (2000), 457–515. (Cited on pages 10, 11, 52, and 58.) [Ca ] D. Calegari, Promoting essential laminations, Invent. Math. (2006), no. 3, 583–643. (Cited on page 45.) [Ca ] D. Calegari, Foliations and the geometry of 3-manifolds.
Oxford Mathematical Mono-graphs. Oxford University Press, Oxford, (2007). xiv+363 pp. ISBN: 978-0-19-857008-0 (Cited on pages 2, 10, 14, and 37.) [CD] D. Calegari, N. Dunfield, Laminations and groups of homeomorphisms of the circle.
In-vent. Math. (2003), no. 1, 149–204. (Cited on page 6.) [Can] A. Candel, Uniformization of surface laminations,
Ann. Sci. ´Ecole Norm. Sup. (1993)489–516. (Cited on page 8.) [CB] A. Casson, S. Bleiler, Automorphisms of surfaces after Nielsen and Thurston.
Lon-don Mathematical Society Student Texts, 9. Cambridge University Press, Cambridge, 1988.iv+105 pp. ISBN: 0-521-34203-1 (Cited on page 58.) [Fen ] S. Fenley, Quasi-isometric foliations. Topology (1992), no. 3, 667–676. (Cited onpage 9.) [Fen ] S. Fenley, Anosov flows in 3-manifolds. Ann. of Math. (2) (1994), no. 1, 79–115. (Cited on pages 6 and 14.) [Fen ] S. Fenley, Foliations, topology and geometry of 3-manifolds: (cid:82) -covered foliations andtransverse pseudo-Anosov flows. Comment. Math. Helv. (2002), no. 3, 415–490. (Citedon pages 10, 11, 17, 52, 53, 54, and 58.) [Fen ] S. Fenley, (cid:82) -covered foliations and transverse pseudo-Anosov flows in atoroidal pieces,arXiv:2101.10984. (Cited on pages 11, 58, 59, and 60.) [FM] S. Fenley, L. Mosher, Quasigeodesic flows in hyperbolic 3-manifolds. Topology (2001),no. 3, 503–537. (Cited on page 43.) [FP] S. Fenley, R. Potrie, Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds, arXiv:1809.02284v2 (Cited on page 8.) [FP ] S. Fenley, R. Potrie, Minimality of the action on the universal circle for uniform foliations.arXiv:2001.05522 (Cited on pages 9, 10, 23, and 46.) [FH] P. Foulon, B. Hasseblatt, Contact Anosov flows on hyperbolic 3-manifolds. Geom. Topol. (2013), no. 2, 1225–1252. (Cited on page 6.) [FrWi] J. Franks and R. Williams Anomalous Anosov flows , in Global Theory of Dyn. Systems,Lecture Notes in Math.
Springer (1980). (Cited on page 5.) [GH] E. Ghys, P. dela Harpe, S ur les groupes Hyperboliques d’apr´es Mikhael Gromov, Progressin Mathematics, Birkh¨auser (1990). (Cited on page 13.) [Go] S. Goodman, Dehn surgery on Anosov flows, Lecture Notes in Math.
Springer (1983)300-307. (Cited on page 6.) [Ha] P. Ha¨ıssinsky, G´eom´etrie quasiconforme, analyse au bord des espaces m´etriques hyper-boliques et rigidit´es, S´eminaire Bourbaki, (2008) 1-39. (Cited on pages 12 and 13.) [HP] A. Hammerlindl, R. Potrie, Partial hyperbolicity and classification: a survey.
ErgodicTheory Dynam. Systems (2018), no. 2, 401–443. (Cited on pages 1, 6, and 7.) [HaPS] A. Hammerlindl, R. Potrie, M. Shannon, Seifert manifolds admitting partially hyper-bolic diffeomorphisms. J. Mod. Dyn. (2018), 193–222. (Cited on pages 1, 57, and 60.) [HaTh] M. Handel, W. Thurston, Anosov flows on new 3-manifolds, Invent. Math. (1980)95-103. (Cited on page 6.) [HPS] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds.
Lecture Notes in Mathematics,
Springer-Verlag, Berlin-New York, 1977. ii+149 pp. (Cited on page 57.) [IM] T. Inaba and S. Matsumoto, Nonsingular expansive flows on 3-manifolds and foliationswith circle prong singularities,
Japan. J. Math. (N.S.) (1990), no. 2, 329–340. (Cited onpages 14 and 45.) [Mar] G. Margulis, Y-flows on three dimensional manifolds; Appendix to ’Certain smooth er-godic systems’ Uspehi Mat. Nauk (1967) no. 5, 107–172.’ by D. Anosov and Y. Sinai. (Cited on pages 2 and 5.) [Mart] S. Martinchich, A note on discretized Anosov flows, In preparation. (Cited on page 57.) [Mos] L. Mosher, Dynamical systems and the homology norm of a 3-mamnifold II,
Invent. Math. (1992) 243-281. (Cited on pages 3 and 52.) [Pat] M. Paternain, Expansive flows and the fundamental group.
Bol. Soc. Brasil. Mat. (N.S.) (1993), no. 2, 179–199. (Cited on pages 14 and 45.) [PT] J. Plante, W. Thurston, Anosov flows and the fundamental group, Topology (1972),147–150. (Cited on page 5.) [Pot] R. Potrie, Robust dynamics, invariant geometric structures and topological classification, Proceedings of ICM
Vol 2 (2018) 2057–2080. (Cited on pages 2 and 6.) [RRS] R. Roberts, J. Shareshian, M. Stein, Infinitely many hyperbolic 3-manifolds which containno Reebless foliation.
J. Amer. Math. Soc. (2003), no. 3, 639–679. (Cited on page 6.) [Sha] M. Shannon, Dehn surgeries and smooth structures on 3-dimensional transitive Anosovflows, PhD Thesis, Dijon (2020). (Cited on pages 3, 5, 14, and 45.) [Th ] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer.Math. Soc. (N.S.) (1988), no. 2, 417–431. (Cited on page 57.) [Th ] W. Thurston, Three-manifolds, Foliations and Circles, I, Preprint arXiv:math/9712268 (Cited on pages 10, 11, 38, and 52.)
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