Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
SSymplectic Geometry of Anosov Flows in Dimension 3 andBi-Contact Topology
Surena HozooriSeptember 24, 2020
Abstract
We give a purely contact and symplectic geometric characterization of Anosov flows in dimen-sion 3 and set up a framework to systematically use tools from contact and symplectic geometryand topology in the study of Anosov dynamics. We also discuss some uniqueness results re-garding the underlying (bi)-contact structures for an Anosov flow and give a characterization ofAnosovity, based on Reeb flows.
Anosov flows were introduced by Dimitri Anosov [1] in 1960s as a generalization of geodesic flowsof hyperbolic manifolds and were immediately considered an important class of dynamical systems,thanks to many interesting global properties. Many tools of dynamical system, including ergodictheory, helped to increase our understanding of Anosov flows (see [37] for early developments).But more profound connections to the topology of the underlying manifold, were discovered indimension 3, thanks to the use of foliation theory. This was initiated by many, including Thurston,Plante and Verjovsky. However, more recent advances in the mid 1990s came from new technicsin foliation theory, introduced by Sergio Fenley, alongside Barbot, Barthelm, etc (see [23] as theseminal work and [6] for a nice survey of such results).The goal of this paper is to set up a new geometric and topological framework for the study ofAnosov flows in dimension 3, thanks to a purely contact and symplectic characterization of suchflows.In this paper, we consider M to be a closed, connected, oriented 3-manifold and assume (projec-tively) Anosov flows to be orientable , i.e. the associated stable and unstable directions are orientableline fields (Assuming the orientability of M , this can be achieved, possibly after going to a doublecover of M ). See Section 2 and Section 3 for related definitions and discussions. Theorem 1.1.
Let φ t be a C flow on the 3-manifold M , generated by the vector field X . Then φ t isAnosov, if and only if, (cid:104) X (cid:105) = ξ + ∩ ξ − , where ξ + and ξ − are transverse positive and negative contactstructures, respectively, and there exist contact forms α + and α − for ξ + and ξ − , respectively, suchthat ( α − , α + ) and ( − α − , α + ) are Liouville pairs. Although the relation to contact geometry was hinted at by Mitsumatsu [41] and Eliashberg-Thurston [20], we use natural geometric quantities, namely growth rates (see Section 3), to turnthose observations into a full characterization. The relation to Mitsumatsu’s work [41] is discussedin the beginning remarks of Section 4. We also note that, although we need to assume C -regularityfor the flows, thanks to structural stability of Anosov flows [1], this does not restrict our study ofmany topological questions, in particular, regarding the orbits of these flows.1 a r X i v : . [ m a t h . S G ] S e p y the above theorem, the vector field, generating an Anosov flow lies in the intersection ofa pair of positive and negative contact structures, i.e. a bi-contact structure . It turns out thatthis condition has dynamical interpretation and defines a large class of flows, named projectivelyAnosov flows (introduced in [41]). These are flows, which induce, via π : T M → T M/ (cid:104) X (cid:105) , a flowwith dominated splitting on T M/ (cid:104) X (cid:105) (see Section 3).We remark that projectively Anosov flows are previously studied in various contexts, underdifferent names. In the geometry and topology literature, beside projectively Anosov flows, theyare referred to as conformally Anosov flows and are studied from the perspectives of foliationtheory [20][42][3], Riemannian geometry of contact structures [9][44][35] and Reeb dynamics [34].This is while, in the dynamical systems literature, the term conformally Anosov is preserved foranother dynamical concept (for instance see [36][48][13]) and the dynamical aspects of projectivelyAnosov flows are studied under the titles flows with dominated splitting (see [11][46][45][47][2]) or eventually relatively pseudo hyperbolic flows [31].Although, it is not immediately clear if the class of projectively Anosov flows is larger thanAnosov flows, first examples of such flows on T and Nil manifolds [41][20], which do not admitany Anosov flows [43], as well as more recent examples of projectively Anosov flows on atoroidalmanifolds, which cannot be deformed to Anosov flows [10], proved the properness of the inclusion.In fact, we now know that unlike Anosov flows, projectively Anosov flows are abundant. Forinstance, there are infinitely many distinct projectively Anosov flows on S and no Anosov flows[4]. Therefore, Theorem 1.1 can be seen as a host of geometric and topological rigidity conditionson a projectively Anosov flow. In particular, this enables us to use various contact and symplecticgeometric and topological tools in the study of Anosov dynamics. For instance, there are manyquestions about the knot theory of the periodic orbits of Anosov flows. Thanks to Theorem 1.1,such periodic orbits are now Legendrian knots for both underlying contact structures and moreover,correspond to exact Lagrangians in the constructed Liouville pairs. These are standard and wellstudied objects in contact and symplectic topology and now, the same technics can be employedfor understanding the periodic orbits of such flows (see Remark 4.10).In contact topology, thanks to
Darboux theorem , there are no local invariants and
Gray’s the-orem implies that homotopy through contact structures can be done by an isotopy of the ambientmanifold. Therefore, the local structure of contact structures does not carry any information andthe subtlety of these structures is hidden in their global topological properties. In fact, we have ahierarchy of topological rigidity conditions on a contact manifold (see Section 2). Although it isnot trivial, it is known that all the inclusions below are proper. (cid:26)
Stein fillablecontact manifolds (cid:27) ⊂ (cid:26) Exactly symplectically fillablecontact manifolds (cid:27) ⊂ (cid:26) Strongly symplectically fillablecontact manifolds (cid:27) ⊂ (cid:26) Weakly symplectically fillablecontact manifolds (cid:27) ⊂ (cid:26) Tightcontact manifolds (cid:27) ⊂ (cid:26) contact manifolds (cid:27) . Now, we can naturally apply the hierarchy of contact topology to bi-contact structures andtherefore, achieve a filtration of Anosovity concepts (see Section 7 for precise definitions). (cid:26)
Anosov flows (cid:27) ⊆ (cid:26) Exactly symplectically bi-fillableprojectively Anosov flows (cid:27) ⊆ (cid:26) Strongly symplectically bi-fillableprojectively Anosov flows (cid:27) ⊂ (cid:26) Weakly symplectically bi-fillableprojectively Anosov flows (cid:27) ⊆ (cid:26) Tightprojectively Anosov flows (cid:27) ⊂ (cid:26) projectively Anosov flows (cid:27) .
2n the above hierarchy, we first notice that there are no equivalent of Stein fillable contactmanifolds for bi-contact structures (and projectively Anosov flows), since Stein fillings can onlyhave connected boundaries [39].The above hierarchy invokes two general lines of questioning, which can help us understandAnosov dynamics, through the lens of contact and symplectic topology. We discuss more precisequestions and conjectures, along these lines, in Section 7.
Question 1.2.
What does each bi-contact topological layer imply about the dynamics of the corre-sponding class of projectively Anosov flows?
Question 1.3.
What bi-contact topological layer is responsible for a given property of Anosovflows?
In this direction, [22] shows that there are no tight projectively Anosov flows on S (generalizingnon-existence of Anosov flows) and [4] gives a partial classification of overtwisted projectively Anosovflows , i.e. when both contact structures, forming the underlying bi-contact structure, are not tight(are overtwisted ). More precisely, they show that overtwisted projectively Anosov flows exist, whenthere are no algebraic obstruction. Although, this is not a full classification, it is worth comparingthis with purely algebraic classification of overtwisted contact structures, by Eliashberg [16][17],confirming the parallels in the two theories. This also implies that the class of tight projectivelyAnosov flows is (considerably) smaller than general projectively Anosov flows.We will prove that properness of the middle inclusion, by constructing examples on T , whilethe properness of other inclusions remain an open problem (Question 7.11). Theorem 1.4.
There are (trivially) weakly symplectically bi-fillable projectively Anosov flows,which are not strongly symplectically bi-fillable.
We finally note that all the above contact topological conditions on projectively Anosov flowsare purely topological, that is do not depend on the homotopy of any of the two underlying contactstructures, with the exception of the first inclusion, i.e. Anosovity of a flow. It turns out that inthe study of bi-contact structures (or equivalently, projectively Anosov flows), the local geometry ismore subtle than contact structures, due to lack of theorems equivalent to the Darboux and Graytheorems. Drawing contrast between two notions of bi-contact homotopy vs. isotopy , we concludethat bi-contact homotopy is the natural notion from dynamical point of view (see Definition 7.1 andthe subsequent discussion). The relation between Anosovity and geometry of bi-contact structuresis not well understood and we bring related discussions and questions in Section 7.
Question 1.5.
How much does the Anosovity of a flow depend on the geometry of the underlyingbi-contact structure, under bi-contact homotopy?
We prove that at least for a fixed projectively Anosov flow, there is a unique supporting bi-contact structure, up to bi-contact homotopy.
Theorem 1.6. If ( ξ − , ξ + ) and ( ξ (cid:48)− , ξ (cid:48) + ) are two supporting bi-contact structures for a projectivelyAnosov flow, then they are homotopic through supporting bi-contact structures. Furthermore, for an Anosov flow, the construction of Liouville pairs in Theorem 1.1 does notdepend on the choice of such bi-contact structure.
Theorem 1.7.
Let φ t be a C Anosov flow on the 3-manifold M , generated by the vector field X and ( ξ − , ξ + ) any supporting bi-contact structure for X . Then there exist negative and positivecontact forms, α − and α + respectively, such that ker α − = ξ − , ker α + = ξ + , and ( α − , α + ) and ( − α − , α + ) are Liouville pairs.
3e also use the literature of Anosov dynamics, as well as the underlying technic of Theorem 1.6,to derive a family of uniqueness results for the underlying contact structures, reducing the studyof the supporting bi-contact structure to only one of the supporting contact structures.
Theorem 1.8. If M is atoroidal and ( ξ − , ξ + ) a supporting bi-contact structure for the Anosovvector field X on M , then for any supporting positive contact structure ξ , ξ is isotopic to ξ + ,through supporting contact structures. Theorem 1.9.
Let X be an R -covered Anosov vector field, supported by the bi-contact structure ( ξ − , ξ + ) on M , and let ξ be any supporting positive contact structure. Then ξ is isotopic to ξ + ,through supporting contact structures. Theorem 1.10.
Let X be the suspension of an Anosov diffeomorphism of torus, supported by thebi-contact structure ( ξ − , ξ + ) , and ξ a positive supporting contact structure. Then, ξ is isotopicthrough supporting bi-contact structures to ξ + , if and only if, ξ is strongly symplectic fillable. On a separate note, we also use the ideas developed in Section 3 and proof of Theorem 1.1 to givea characterization of Anosovity, based on the
Reeb vector fields , associated to the underlying contactstructures. Reeb vector fields play a very important role in contact geometry and Hamiltonianmechanics and since early 90s, their deep relation to the topology of contact manifolds has beenexplored.
Theorem 1.11.
Let X be a projectively Anosov vector field on M . Then, the followings areequivalent:(1) X is Anosov;(2) There exists a supporting bi-contact structure ( ξ − , ξ + ) , such that ξ + admits a Reeb vectorfield, which is dynamically negative everywhere;(3) There exists a supporting bi-contact structure ( ξ − , ξ + ) , such that ξ − admits a Reeb vectorfield, which is dynamically positive everywhere. In Section 2, we review some backgrounds from contact and symplectic topology, which providecontext for this paper. In Section 3, we discuss (projective) Anosovity of flows, with emphasis on thegeometry of the growth in the stable and unstable direction, which lies in the heart of Theorem 1.1.We prove Theorem 1.1 and 1.7 in Section 4. Section 5 is devoted to various uniqueness theoremsfor the underlying (bi)-contact structures of (projectively) Anosov flows. In Section 6, we proveTheorem 1.11, as a consequence of ideas developed in Section 3 and the proof of Theorem 1.1.Finally, Section 7 is devoted to setting up the contact topological framework for the systematic useof contact and symplectic topological methods in Anosov dynamics. The necessary remarks anddefinitions are discussed there, as well as a handful of related open problems and conjectures. Aproof of Theorem 1.4 is also given in this section.
ACKNOWLEDGEMENT:
I want to thank my advisor, John Etnyre, for continuous supportand discussions along the way, as well as suggesting Example 7.7 and helping with the details of theproof of Theorem 5.7. I am also very thankful to Rafael de la Llave, Gabriel Paternain, JonathanBowden and Thomas Barthelm for helpful discussions and suggested revisions. The author waspartially supported by the NSF grants DMS-1608684 and DMS-1906414.
In this section, we review some basic notions from contact and symplectic geometry and topology,which will be useful in the rest of the paper. We refer the reader to [28] for more on these topics.4 efinition 2.1.
We call the 1-form α a contact form on M , if α ∧ dα is a non-vanishing volumeform on M . If α ∧ dα > (compared to the orientation on M ), we call α a positive contact formand otherwise, a negative one. We call ξ := ker α a (positive or negative) contact structure on M .Moreover, we call the pair ( M, ξ ) a contact manifold . When not mentioned, we assume the contactstructures to be positive. Recall that by Frobenius theorem, the contact structure ξ in the above definition is a coorientablemaximally non-integrable plane field on M . Example 2.2.
Some examples of contact structures are:1) The 1-form α std = dz − y dx is a positive contact form on R . We call ξ std = kerα std the standard positive contact structure on R . Similarly, ker ( dz + y dx ) is the standard negativecontact structure on R .2) Consider C equipped with J , the standard complex structure on T C and let S be the unitsphere in C . It can be seen that the plane field ξ std := T S ∩ J T S is a contact structure on S ,referred to as standard contact structure on S . Alternatively, ξ std can be defined as the uniquecomplex line tangent to the unit sphere. It is helpful top note that this contact structure is theone point compactification of the standard contact structure on R . Similarly, we can construct anegative contact structure on S , by considering the conjugate of the complex structure J .3) Consider T (cid:39) R / Z . It can be seem that the plane fields ξ n = ker { cos 2 πnzdx − sin 2 πnzdy } are positive and negative contact structures on T , for integers n > and n < , respectively.Gray’s theorem states that members of any C -family of contact structures are isotopic as con-tact structures and according to Darboux theorem , all contact structures locally look the same. i.e.around each point in a contact manifold (
M, ξ ), there exists a neighborhood U and a diffeomor-phism of U to R , mapping ξ to the standard contact structure (positive or negative one, dependingon whether ξ is positive or negative) on R . While this means that contact structures lack localinvariants, it turns out understanding their topological properties is more subtle and interesting.The most important global feature of contact structures is tightness , introduced by Eliashberg [16],and determining whether a given contact is tight, as well as classifying such contact manifolds, area prominent theme in contact topology. Definition 2.3.
The contact manifold ( M, ξ ) is called overtwisted , if M contains an embedded diskthat is tangent to ξ along its boundary. Otherwise, ( M, ξ ) is called tight . Moreover, ξ universallytight, if its lift to the universal cover of M is tight as well. The significance of the above dichotomy is the classification of overtwisted contact structures byY. Eliashberg [17][16]. He showed overtwisted contact structures, up to isotopy, are in one to onecorrespondence with plane fields, up to homotopy (in particular, they always exist). This meansovertwisted contact structures do not carry more topological information than plane fields. On theother hand, tight contact structures reveal deeper information about their underlying manifold,and are harder to find, understand and classify.
Remark 2.4.
It can be shown that all the contact structures in Example 2.2 are (universally) tight.In fact, they are the only tight contact structures, up to isotopy, on their underlying manifolds. Note,that all those manifolds admit overtwisted contact structures as well.
It turns out that one can determine tightness of a contact manifold is based on its relation tofour dimensional symplectic topology , the even dimensional sibling of contact topology.
Definition 2.5.
Let X be an oriented 4-manifold. We call a 2-form ω on X a symplectic form, ifit is closed and ω ∧ ω > . The pair ( X, ω ) is called a symplectic manifold. ω std = d ( x dy + x dy ) = dx ∧ dy + dx ∧ dy is a symplectic form definedon R with coordinates ( x , y , x , y ) (known as the standard symplectic form), and the Darbouxtheorem in symplectic geometry states that all symplectic structures are locally equivalent, up to symplectic deformation . Using the theory of J -holomorphic curves, Gromov and Eliashberg proved[30][18] that a contact structure is tight, when it is symplectically fillable , even in the weakest sense. Definition 2.6.
Let ( M, ξ ) be a contact manifold. We call the symplectic manifold ( X, ω ) a weaksymplectic filling for ( M, ξ ) , if ∂X = M as oriented manifolds and ω | ξ > . We call ( X, ω ) a strongsymplectic filling , if moreover, ω = dα in a neighborhood of M = ∂X , for some 1-form α , suchthat α | T M is a contact form for ξ . Finally, we call ( X, ω ) an exact symplectic filling for ( M, ξ ) ,if such 1-form α can be defined on all of X . We call such ( M, ξ ) (weakly, strongly or exactly)symplectically fillable . Theorem 2.7. (Gromov 85 [30], Eliashberg 90 [18]) If ( M, ξ ) is (weakly, strongly or exactly)symplectically fillable, then it is tight. Remark 2.8.
We note that for a 2-form ω to be symplectic, it needs to be at least C , because ofthe closedness condition. However, when ω is exact, i.e. ω = dα for some 1-form α , this conditionis automatically satisfied, assuming the required regularity. So for most purposes, we don’t needto assume, for an exact 2-form ω = dα , any regularity more than C , and methods of symplecticgeometry and topology, in particular, the use of J -holomorphic curves and Theorem 2.7, can beapplied. We can also approximate such ω = dα , by symplectic forms of arbitrary high regularity,using C -approximations of α . Therefore, we still call such ω symplectic, especially in Theorem 4.1. It is known that not all tight contact structures are weakly symplectically fillable, the set ofstrongly symplectically fillable contact manifolds is a proper subset of the set of weakly symplec-tically fillable contact manifolds and the set of exactly symplectically fillable contact manifoldsis a proper subset of the set of strongly symplectically fillable contact manifolds. Moreover, if adisconnected contact manifold is strongly or weakly symplectically fillable, each of its componentsis strongly or weakly symplectically fillable, respectively [15][21].
Example 2.9.
1) The unit ball in ( R , ω std ) is a strong symplectic filling for ( S , ξ std ) , consideredas the unit sphere in R .2) We want to show that all tight contact structures on T , given in Example 2.2 3), are weaklysymplectically fillable. We can observe that after an isotopy ξ n = ker dz + (cid:15) { cos 2 πnzdx − sin 2 πnzdy } for small (cid:15) > , i.e. we can isotope ξ n to be arbitrary close to the horizontal foliation ker dz on T . Now consider the symplectic manifold ( X, ω ) = ( T × D , ω ⊕ ω ) , where ω and ω are areaforms for T and D , respectively. Clearly, ∂X = T and if at the boundary, we consider thecoordinates ( x, y ) for T and z for the angular coordinate of D , we have ω | ker dz > . Since, forsmall (cid:15) > , ξ n is a small perturbation of ker dz , we also have ω | ξ n > . Therefore, all ξ n s areweakly symplectically fillable. It can be seen [19] that except ξ , none of these contact structures,are strongly symplectically fillable and the canonical symplectic structure on the cotangent bundle T ∗ T , provides an exact symplectic filling for ( T , ξ ) . Remark 2.10.
The concept of
Giroux torsion was introduced by Emmanuel Giroux [29]. A contactmanifold ( M, ξ ) is said to contain Giroux torsion, if it admits a contact embedding of (cid:0) [0 , π ] × S × S with coordinates ( t, φ , φ ) , ker (cos t dφ + sint dφ ) (cid:1) → ( M, ξ ) . Note that all the tight contact structures on T , discussed in Example 2.2 3) contain Girouxtorsion, except for n = 1 . Later in [27], it was proven that contact structures containing Giroux orsion do not admit strong symplectic fillings. This notion can be generalized by considering Giroux π -torsion . i.e. when the contact manifold contains half of a Giroux torsion: (cid:0) [0 , π ] × S × S with coordinates ( t, φ , φ ) , ker cos t dφ + sint dφ (cid:1) → ( M, ξ ) . In Example 2.2 3), for all n > , the contact manifold ( T , ξ n ) contains Giroux torsion, while ( T , ξ ) is constructed by gluing two Giroux π -torsions along their boundary. An special case of exact symplectic fillings was observed by Mitsumatsu, in the presence ofsmooth volume preserving Anosov flows [41] (see Section 4 for more discussion and improvementof Mitsumatsu’s theorem). Alongside [39], these were the first examples of exact symplectic fillingswith disconnected boundaries. Explicit examples of such structures can be found in [41] and theyare also discussed in [38].
Definition 2.11.
We call a pair ( α − , α + ) a Liouville pair , if α − and α + are negative and positivecontact forms, respectively, whose kernels are transverse and M × [ − , t , equipped with the symplec-tic structure d { (1 − t ) α − + (1 + t ) α + } ) is an exact symplectic filling for ( M, ker α + ) (cid:116) ( − M, ker α − ) ,where − M is M with reversed orientation. In this section, we review the basic facts about Anosovity, emphasizing on the growth behavior ofthe flows in stable and unstable directions, from a geometric point of view.In the following, we assume X is a non-zero C -vector field on a closed, oriented 3-manifold M . Definition 3.1.
We call the C flow φ t Anosov , if there exists a splitting
T M = E ss ⊕ E uu ⊕ (cid:104) X (cid:105) ,such that the splitting is continuous and invariant under φ t ∗ and || φ t ∗ ( v ) || ≥ Ae Ct || v || for any v ∈ E uu , || φ t ∗ ( u ) || ≤ Ae − Ct || u || for any u ∈ E ss , where C and A are positive constants, and || . || is induced from some Riemannian metric on T M .We call E uu and E ss , the strong unstable and stable directions, respectively. Moreover, we call thevector field X , the generator of such flow, an Anosov vector field . In this paper, we assume E ss and E uu to be orientable. This can be arranged, possibly aftergoing to a double cover of M . Example 3.2.
Classic examples of Anosov flows in dimension 3 include the geodesic flows on theunit tangent space of hyperbolic surfaces and suspension of Anosov diffeomorphisms of torus. Bynow, we know that there are Anosov flows on hyperbolic manifolds as well [26].
Here, we note that by [1], a small perturbation of any Anosov flow is Anosov and moreover,is orbit equivalent to the original flow, i.e. there exists a homeomorphism mapping the orbits ofthe perturbed flow to the orbits of the original flow. Therefore, for most practical purposes we canassume higher regularity for the flow. For our purposes, it suffices for the flow to be C (or thegenerating vector field to be C ).In [41] and [20], it is shown that C Anosov vector fields span the intersection of a pair oftransverse positive and negative contact structures, i.e. a bi-contact structure . However, it isknown that the inverse is not true. As a matter of fact, non-zero vector fields in the intersection ofa bi-contact structure define a considerably larger class of vector fields, namely projectively Anosovvector fields . By [41] and [20], this is equivalent to the following definition (note that since contactstructures are at least C , the flow needs to be at least C ):7 − ξ + E u E s (a) Anosov flows E u E s ξ − ξ + (b) Projectively Anosov flows Figure 1: The local behavior of (projectively) Anosov flows
Definition 3.3.
We call a C flow φ t projectively Anosov , if its induced flow on T M/ (cid:104) X (cid:105) admitsa dominated splitting . That is, there exists a splitting T M/ (cid:104) X (cid:105) = E s ⊕ E u , such that the splittingis continuous and invariant under ˜ φ t ∗ and || ˜ φ t ∗ ( v ) || / || ˜ φ t ∗ ( u ) || ≥ Ae Ct || v || / || u || for any v ∈ E u ( unstable direction ) and u ∈ E s ( stable direction ), where C and A are positiveconstants, X is the generator of the flow φ t , || . || is induced from some Riemannian metric on T M/ (cid:104) X (cid:105) and ˜ φ t ∗ is the flow induced on T M/ (cid:104) X (cid:105) .Moreover, we call the vector field X , a projectively Anosov vector field . Similar to Anosov flows, we assume the orientability of the stable and unstable directions ofprojectively Anosov flows in this paper.In [41] and [20], it is shown:
Proposition 3.4.
Let X be a C vector field on M . Then, X is projectively Anosov, if and onlyif, there exist positive and negative contact structures, ξ + and ξ − respectively, which are transverseand X ⊂ ξ + ∩ ξ − . This motivates the following definitions:
Definition 3.5.
We call the pair ( ξ − , ξ + ) a bi-contact structure on M , if ξ + and ξ − are positiveand negative contact structures on M , respectively, and ξ − (cid:116) ξ + . Definition 3.6.
Let X be a projectively Anosov vector field on M . We call a bi-contact structure ( ξ − , ξ + ) a supporting bi-contact structure for X , or the generated projectively Anosov flow, if X ⊂ ξ − ∩ ξ + . We call a positive (negative) contact structure or more generally, any plane field ξ , asupporting positive (negative) contact structure or plane field, respectively, for X or the generatedflow, if X ⊂ ξ . See Example 7.7 for explicit examples of projectively Anosov flows (which are not Anosov).Similar examples can be constructed on Nil manifolds as well [41]. Also, see [10] for using theidea of hyperbolic plugs [8] to construct of projectively Anosov flows (which cannot be deformed,through projectively Anosov flows, into Anosov flows) on atoridal manifolds.Consider the vector bundle π : T M → T M/ (cid:104) X (cid:105) and notice that for any plane field η which istransverse to the flow, there exists a natural vector bundle isomorphism T M/ (cid:104) X (cid:105) (cid:39) η , induced byprojection onto η and along X . Therefore, π can be interpreted as such projection as well.8e also notice that in Definition 3.3, the line fields E u , E s ⊂ T M/ (cid:104) X (cid:105) do not necessarily lift toinvariant line fields E uu , E ss ⊂ T M , respectively (see [42] for the examples of when they do not).However, it is a classical fact from dynamical systems that when the induced flow on
T M/ (cid:104) X (cid:105) (usually called Linear Poincar Flow ) admits an invariant continuous hyperbolic splitting E s ⊕ E u (uniformly contracting along E s and expanding along E u ), such lift does exist and we we will havean invariant splitting as in Definition 3.1 (see [14], Proposition 1.1). Definition 3.7.
We call a projectively Anosov flow (vector field) balanced , if it preserves a trans-verse plane field η . Proposition 3.8.
The C flow φ t is a balanced projectively Anosov, if and only if, there exists asplitting T M = E ss ⊕ E uu ⊕ (cid:104) X (cid:105) , such that the splitting is continuous and invariant under φ t ∗ and || φ t ∗ ( v ) || / || φ t ∗ ( u ) || ≥ Ae Ct || v || / || u || for any v ∈ E uu ( strong unstable direction ) and u ∈ E ss ( strong stable direction ), where C and A are positive constants, and X is the generator of the flow.Proof. We can easily observe E ss = η ∩ π − ( E s ) and E uu = η ∩ π − ( E u ). Remark 3.9.
It can be seen that given a projectively Anosov flows, the plane fields π − ( E s ) and π − ( E u ) are C integrable plane fields, named stable and unstable foliations , respectively. In theAnosov case, thanks to ergodic theory, higher regularity of these foliations can be assumed and thatis the basis for the use of foliation theory to study Anosov dynamics. Therefore, such tools are notall well transferred to projectively Anosov dynamics in general. However, assuming more regularityfor the associated foliations of a projectively Anosov flow, some rigidity results are known [42][3]. It is worth to pause and make few observations about the geometry of projectively Anosov flows(the remark is discussed more in depth in [20]).
Remark 3.10. If X is some vector field on M , which is tangent to some plane field ξ , we canmeasure the contactness of ξ from the rotation of the flow, with respect to ξ , in the following way.Choose some transverse plane field η , which is differentiable in the direction of X (for instance, if X is a balanced projectively Anosov vector field, E ss ⊕ E uu can be chosen) and orient it such that X and η induce the chosen orientation of M . Let λ = ξ ∩ η and λ tp = φ − t ∗ ( ξ φ t ( p ) ) ∩ η for x ∈ M and t ∈ R . Finally, let θ tp be the angle between λ p and λ p ( t ) , for some Riemannian metric, whichis differentiable in the direction of X . Then, ξ is a positive or negative contact structure, if andonly if, X · θ p ( t ) < or X · θ p ( t ) > , respectively, for all p and t .Now, if X is a projectively Anosov vector field, and ( ξ − , ξ + ) a bi-contact structure such that X ⊂ ξ − ∩ ξ + , let η be any transverse plane field, and λ + = ξ + ∩ η and λ − = ξ − ∩ η . Similar toabove, we can define λ t + ,p and λ t − ,p and observe lim t → + ∞ λ t + ,p = lim t → + ∞ λ t − ,p = π − ( E s ) ∩ η and lim t →−∞ λ t − ,p = lim t →−∞ λ t + ,p = π − ( E u ) ∩ η, Equivalently, t → + ∞ φ t ∗ ( ξ + ) = lim t → + ∞ φ t ∗ ( ξ − ) = π − ( E u ) and lim t →−∞ φ t ∗ ( ξ + ) = lim t →−∞ φ t ∗ ( ξ − ) = π − ( E s ) . It turns out that we can characterize Anosovity of a projectively Anosov vector field by the growth of its stable and unstable directions .We first note that the norm used in the definition of a (projectively) Anosov flow X is ingeneral induced from some C Riemannian structure g . However, if we replace g with T (cid:82) T φ t ∗ gdt ,where φ t is the flow of X , the resulting Riemannian metric will be differentiable in X -direction, i.e. L X g T would exist. Moreover, by considering large enough T , with respect to such metric, we canassume A = 1 in the above definitions, meaning that the growth or decay in stable and unstabledirections, respectively, for an Anosov flow, or the relative growth for a projectively Anosov flow,start immediately. Assuming such conditions, we can compute the infinitesimal rate of growthfor vectors in the stable and unstable directions. Remember that any transverse plane field η induces a vector bundle isomorphism T M/ (cid:104) X (cid:105) . Using such isomorphism, the restriction g | η of anyRiemannian metric g on T M , defines a metric on
T M/ (cid:104) X (cid:105) , and conversely, given any Riemannianmetric on T M/ (cid:104) X (cid:105) , we can define a metric on T M , whose restriction on η is induced from suchmetric.Let ˆ e u ∈ E u ⊂ T M/ (cid:104) X (cid:105) be the unit vector field (with respect to some Riemannian metric)defined in the neighborhood of a point. Noticing that the linear flow on T M/ (cid:104) X (cid:105) preserves thedirection of ˆ e u , we compute: L X ˜ e u = ∂∂t ˜ φ − t ∗ (˜ e u ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ∂∂t ˜ φ − t ∗ (cid:16) ˜ φ t ∗ (˜ e u ) (cid:17) || ˜ φ t ∗ (˜ e u ) || (cid:12)(cid:12)(cid:12)(cid:12) t =0 = (cid:18) ∂∂t || ˜ φ t ∗ (˜ e u ) || (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t =0 ˜ e u = − (cid:18) ∂∂t || ˜ φ t ∗ (˜ e u ) || (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t =0 ˜ e u = − (cid:18) ∂∂t ln || ˜ φ t ∗ (˜ e u ) || (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t =0 ˜ e u . We can do similar computation for the (locally defined) unit vector field ˜ e s ∈ E s . Definition 3.11.
Using the above notation, we define the growth rate of (un)stable direction as r s := ∂∂t ln || ˜ φ t ∗ (˜ e s ) || (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:18) r u := ∂∂t ln || ˜ φ t ∗ (˜ e u ) || (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:19) . Proposition 3.12.
The above computation shows: L X ˜ e s = − r s ˜ e s ( L X ˜ e u = − r u ˜ e u ) , and ˜ φ T ∗ (˜ e s ) = e (cid:82) T r s ( t ) dt ˜ e s (cid:16) ˜ φ T ∗ (˜ e u ) = e (cid:82) T r u ( t ) dt ˜ e u (cid:17) . Now consider a transverse plane field η (cid:39) T M/ (cid:104) X (cid:105) , equipped with a Riemannian metric ˜ g on T M/ (cid:104) X (cid:105) , defining stable and unstable growth rate of r s , r u . From ˜ g , a Riemannian metric isinduced on η , which can be extended to a Riemannian metric g on T M , such that L X g exists,assuming that L X ˆ g and L X η exist. Let e s , e u ∈ η be chosen such that π ( e s ) = ˜ e s and π ( e u ) = ˜ e u ,and notice that || e s || = || e u || = 1. Let π η be the projection onto η and along X and compute L X e u = ∂∂t φ − t ∗ ( e u ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ∂∂t π η (cid:0) φ − t ∗ ( e u ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t =0 + q ηu X, q ηu : M → R . Since the metric on η is induced from ˆ g , this implies L X e u = − r u e u + q ηu X. Similarly, L X e s = − r s e s + q ηs X, for some function q ηs : M → R . We have proved: Proposition 3.13.
Let X be a projectively Anosov vector field with r s , r u being its growth rate ofstable and unstable directions (with respect to some metric on T M/ (cid:104) X (cid:105) ), respectively. Then, for anytransverse plane field η , there exists a metric on T M such that for unit vector fields e u ∈ η ∩ π − ( E u ) and e s ∈ η ∩ π − ( E s ) , we have L X e u = − r u e u + q ηu X, and L X e s = − r u e s + q ηs X, for appropriate real functions q ηu , q ηs : M → R . We also observe the following fact, which we will use in the proof of Theorem 4.1:
Proposition 3.14.
Let X be a projectively Anosov vector field. When X is balanced (in particular,when X is Anosov), there exists a transverse plane field η as in Proposition 3.13, for which q ηu = q ηs = 0 everywhere. In this case, L X e s = − r s e s ( L X e u = − r u e u ) , and φ T ∗ ( e s ) = e (cid:82) T r s ( t ) dt e s (cid:16) φ T ∗ ( e u ) = e (cid:82) T r u ( t ) dt e u (cid:17) . The definition of projectively Anosov vector fields implies:
Proposition 3.15.
Let X be a projectively Anosov vector field and r s and r u , the growth rates ofstable and unstable directions, respectively, with respect to any Riemannian metric, satisfying themetric condition of Definition 3.3 with A = 1 , which is differentiable in X -direction, then r u − r s > . Proof.
Since X is projectively Anosov, the exists a Riemannian metric g , such that L X g exists and || ˜ φ t ∗ (˜ e u ) || / || ˜ φ t ∗ (˜ e s ) || ≥ e Ct || ˜ e u || / || ˜ e s || where ˜ φ t is the flow of X , || . || is the norm on T M/ (cid:104) X (cid:105) , induced from g , ˜ e u ∈ E u and ˜ e s ∈ E s areunit vectors, and C is a positive constants. Therefore,ln || ˜ φ t ∗ (˜ e u ) || − ln || ˜ φ t ∗ (˜ e s ) || ≥ Ct and r u − r s = ∂∂t ln || ˜ φ t ∗ (˜ e u ) || (cid:12)(cid:12)(cid:12)(cid:12) t =0 − ∂∂t ln || ˜ φ t ∗ (˜ e s ) || (cid:12)(cid:12)(cid:12)(cid:12) t =0 ≥ C > . emark 3.16. In proof of Theorem 4.1, we will also see that inverse of the above proposition alsoholds, in the sense that given a C projectively Anosov vector field, for any Riemannian metricwith r u − r s > , the plane fields (cid:104) X, π − ( e u + e s ) (cid:105) and (cid:104) X, π − ( e u − e s ) (cid:105) define positive and negativecontact structures, respectively, possibly after a perturbation to make the plane fields C . Similar computation, using the definition of Anosov flows and the fact that hyperbolicity of
T M/ (cid:104) X (cid:105) implies Anosovity of the flow ([14], Proposition 1.1), yield: Proposition 3.17.
Let X be a projectively Anosov vector field and r s and r u . Then X is Anosov,if and only if, with respect to some Riemannian metric, we have r u > > r s . Remark 3.18.
The above computation also shows that both Anosovity and projective Anosovity arepreserved under reparametrizations of the flow. More precisely, let X is projectively Anosov vectorfield with growth rates of r s and r u , in the stable and unstable directions, respectively (with respectto some metric). Then, for any positive function f : M → R > , the vector field f X has growthrates of f r s and f r u , in the stable and unstable directions, respectively (with respect to the samemetric). Therefore, the conditions of both Proposition 3.15 and Proposition 3.17, are preservedunder such transformations. In [41], Mitsumatsu showed that the generator vector field of any smooth volume preserving Anosovflow lies in the intersection of a pair of transverse negative and positive contact structures, admittingcontact forms α − and α + , respectively, such that ( α − , α + ) is a Liouville pair (see Definition 2.11).He used the volume preserving property, to show that the linear combination of α − and α + issymplectic (in the sense of Definition 2.11) and used the smoothness of the flow to be able toconsider C -regularity for the leaves of stable and unstable foliations, used in constructing thedesired geometric structures. In the following, we improve these results by showing the same holdswithout those assumption, as well proving the converse, achieving a full characterization of C Anosov flows in terms of contact and symplectic geometry. We carefully use the growth rate ofstable and unstable directions (see Section 3) to get the non-degeneracy of the symplectic form,without using a preserved volume. Furthermore, we use various approximations of the stable andunstable foliations, instead of regularity assumptions. It is worth noting that since the generatingvector field is C and the metric used in the definition of (projectively) Anosov flows can be chosento be differentiable in the direction of the flow, all the geometric objects and quantities we careabout, like stable and unstable directions, growth rates, etc. behave well in the direction of theflow. The subtlety is to approximate the desired geometric structures by structures which are C ,while preserving the desired quantities, which are based on differentiation in the direction of theflow. Theorem 4.1.
Let φ t be a C flow on the 3-manifold M , generated by the vector field X . Then φ t isAnosov, if and only if, (cid:104) X (cid:105) = ξ + ∩ ξ − , where ξ + and ξ − are transverse positive and negative contactstructures, respectively, and there exist contact forms α + and α − for ξ + and ξ − , respectively, suchthat ( α − , α + ) and ( − α − , α + ) are Liouville pairs.Proof. We begin by assuming φ t is Anosov. 12 − ξ + E u E s e s e u Figure 2:
T M/ (cid:104) X (cid:105) (cid:39) E s ⊕ E u Let g be the C Riemannian metric for which the condition of Anosovity is satisfied and g ( X, E ss ) = g ( X, E uu ) = g ( E ss , E uu ) = 0. After replacing g with T (cid:82) T φ t ∗ g ( t ) dt for large T ,we can assume the same orthogonality conditions hold, L X g exists everywhere and the growth anddecay of E u u and E ss start immediately (i.e. can assume A = 1 in the Definition 3.1). This meansthat if e s ∈ E ss and e u ∈ E uu are unit vector fields, the stable and unstable growth rates, r s and r u are defined and are negative and positive, respectively (Proposition 3.17). Moreover, choose such e s and e u so that ( e s , e u , X ) is an oriented basis for M as in Figure (2).Let α preu and α pres be C -approximations for ˆ e u and ˆ e s , g -duals of e u and e s , respectively, suchthat α preu ( X ) = α pres ( X ) = 0. To do so, we need to C -approximate E s and E u as line bundles in T M/ (cid:104) X (cid:105) . The direct sum of such line bundles with (cid:104) X (cid:105) yields the desired C plane fields.There exist continuous functions f u and f s which are differentiable in direction of X and f u α preu ( e u ) = f s α pres ( e s ) = 1. By [25], we can C -approximate f u and f s with smooth functions ˜ f u and ˜ f s , such that | X · f u − X · ˜ f u | and | X · f s − X · ˜ f s | are arbitrary small. In particular, we canfind such ˜ f u and ˜ f s for which X · [ ˜ f u α preu ( e u )] + (min x ∈ M r u ) ˜ f u α preu ( e u ) > X · [ ˜ f s α pres ( e s )] + (max x ∈ M r s ) ˜ f s α pres ( e s ) < . (2)Define α u := ˜ f u α preu and α s := ˜ f s α pres .In the following, when there is no confusion, for any point x ∈ M , we refer to r s ( x ) by r s or r s (0) and to r s ( φ t ( x )) by r s ( t ). Similarly, for other functions in this proof.Now define α Tu := I Tu φ T ∗ α u , α Ts := I − Ts φ − T ∗ α s ;where I Tu := e − (cid:82) T r u ( t ) dt ,I Ts := e − (cid:82) T r s ( t ) dt . Claim 4.2. α Tu ( e u (0)) = α u ( e u ( T )) ,α Ts ( e s (0)) = α s ( e s ( T )) . roof. α Tu ( e u (0)) = I Tu α u ( φ T ∗ ( e u (0))) = I Tu α u ( 1 I Tu e u ( T )) = α u ( e u ( T )) , where the middle equality is implied by Proposition 3.14. Other implication follows similarly. Claim 4.3. lim T → + ∞ I Tu I Ts = lim T → + ∞ I − Ts I − Tu = 0 . Proof. lim T → + ∞ I Tu I Ts = lim T → + ∞ e (cid:82) T r s ( t ) − r u ( t ) dt = 0 . The last equality follows from projective Anosovity of X (Proposition 3.15), implying r u − r s > T → + ∞ I − Ts I − Tu = lim T → + ∞ e (cid:82) − T r s ( t ) − r u ( t ) dt = 0 . . Claim 4.4. X · I Tu = [ r u (0) − r u ( T )] I Tu ; X · I Ts = [ r s (0) − r s ( T )] I Ts . Proof. X · I Tu = ∂∂h e − (cid:82) T r u ( t + h ) dt (cid:12)(cid:12)(cid:12)(cid:12) h =0 = [ − (cid:90) T r (cid:48) u ( t ) dt ] I Tu = [ r u (0) − r u ( T )] I Tu . The other implication follows similarly.Now, using the above calculations, we can show that ker α Tu and ker α Ts , C -converge to π − ( E s ) = E ss ⊕ (cid:104) X (cid:105) and π − ( E u ) = E uu ⊕ (cid:104) X (cid:105) , respecting certain C -quantities. Lemma 4.5.
We have lim T → + ∞ ker α Tu = π − ( E s ) and lim T → + ∞ ker α Ts = π − ( E u ) . Proof.
First computelim T → + ∞ α Tu ( e s (0)) = lim T → + ∞ I Tu α u ( φ T ∗ e s (0)) = lim T → + ∞ I Tu I Ts α u ( e s ( T )) = 0 . The last equality follows from Claim 4.3 and the fact that α u ( e s ) is bounded. Similarly,lim T → + ∞ α Ts ( e u (0)) = 0 . Claim 4.2 and the fact that α Tu ( X ) = α Ts ( X ) = 0 finish the proof.14ow, we see that certain C -variations behave nicely under such limiting procedure. Lemma 4.6. lim T → + ∞ α Tu ∧ dα Tu = lim T → + ∞ α Ts ∧ dα Ts = 0 . Proof.
Using Claim 4.2, Claim 4.3 and Claim 4.4, compute( α Tu ∧ dα Tu )( e s , e u , X ) = α Tu ( e s ) (cid:2) − X · ( α Tu ( e u )) + α Tu ( L X e u ) (cid:3) − α Tu ( e u ) (cid:2) X · ( α Tu ( e s )) − α Tu ( L X e s ) (cid:3) = α u ( e s ( T )) (cid:2) − X · ( α u ( e u ( T ))) − r u α u ( e u ( T )) (cid:3) I Tu I Ts − α u ( e u ( T )) (cid:2) ( r u (0) − r u ( T ) − r s (0) + r s ( T )) α u ( e s ( T )) + X · ( α u ( e s ( T ))) + r s α u ( e s ( T )) (cid:3) I Tu I Ts = A ( x ) I Tu I Ts , where A ( x ) is a bounded function on M .Claim 4.3 concludes the implication and similar computation for α Ts ∧ dα Ts finishes the proof. Remark 4.7.
Using Proposition 3.13, one can easily check that Claim 4.2, 4.3, 4.4 and Lemma 4.5,4.6 also hold for similar approximations, when the flow is merely projectively Anosov.
Lemma 4.8.
For large enough T , α Tu ∧ dα Ts and α Ts ∧ dα Tu are negatively bounded away from .Proof. ( α Tu ∧ dα Ts )( e s , e u , X ) = α Tu ( e s )[ − X · ( α Ts ( e u )) − α Ts ( −L X e u )] + α Tu ( e u )[ X · ( α Ts ( e s )) − α Ts ( L X e s )]= I Tu I Ts I − Ts I − Tu A ( x ) + α u ( e u ( T ))[ X · ( α s ( e s ( T ))) + r s α s ( e s ( T ))] , where A ( x ) is a bounded function on M . Using Claim 4.3, the first term vanishes in the limit andwe will have α Tu ∧ dα Ts < , since by criteria (2) we forced the second term to be negatively bounded away from 0.Similar computation and criteria (1) implies the other statement.Now we have all the ingredients to finish the proof.Let α T + := ( α Tu − α Ts ) and α T − := ( α Tu + α Ts ). The goal is to show that ( α T − , α T + ) and ( − α T − , α T + )are Liouville pairs, for large T . By Lemma 4.6 and 4.8, for large T : α T + ∧ dα T + = 14 (cid:0) α Tu ∧ dα Tu − α Tu ∧ dα Ts − α Ts ∧ dα Tu + α Ts ∧ dα Ts (cid:1) > . Therefore, α T + is a positive contact form for large T . Similar computation shows that α T − is anegative contact form for large T .To show that ( α T − , α T + ) is a Liouville pair, we need to show that ω T := dα T is a symplectic formon M × [ − , α T := { α Tt } t ∈ [ − , and α Tt := (1 − t ) α T − + (1 + t ) α T + = α Tu − tα Ts .Compute for large T : ω T ∧ ω T = ( dα Tu − tdα Ts − dt ∧ α Ts ) ∧ ( dα Tu − tdα Ts − dt ∧ α Ts ) =15 dt ∧ {− α Ts ∧ dα Tu + 2 tα Ts ∧ dα Ts } > . Then, Lemma 4.6 and 4.8 imply that ω T is symplectic for large T .To show that ( − α T − , α T + ) is a Liouville pair, let ˜ ω T := d ˜ α T , where ˜ α T := { ˜ α Tt } t ∈ [ − , and˜ α Tt := − (1 − t ) α T − + (1 + t ) α T + = α Ts − tα Tu . Similar computation shows:˜ ω T ∧ ˜ ω T = dt ∧ {− α Tu ∧ dα Ts + 2 tα Tu ∧ dα Tu } > , implying that ˜ ω T is symplectic for large T and finishing the proof of one implication.We now consider the other implication.Note that by Proposition 3.4, such flow is projectively Anosov and therefore, we have thesplitting T M/ (cid:104) X (cid:105) (cid:39) E s ⊕ E u . Without loss of generality, assume α + and α − induce the same ori-entation on π − ( E u ) and opposite orientations on π − ( E s ) (recall that π is the fiberwise projection T M → T M/ (cid:104) X (cid:105) ). The idea is to show that for any point in M , when constructing the Liouvilleform by linearly interpolating α + and α − (or − α − ), the symplectic condition at the time , whenthe kernel of the interpolation is π − ( E s ) (or π − ( E u )), implies r u > r s < time is a continuous function on the manifold. But it turns out that, thanks to the openness of thesymplectic condition, suitable approximation by a C function suffices.Orient π − ( E u ) such that α + ( π − ( E u )) > α − ( π − ( E u )) >
0. Also orient π − ( E s ) suchthat α + ( π − ( E s )) < < α − ( π − ( E s )).Let τ u ( x ) be the (continuous) function such thatker { (1 − τ u ) α − + (1 + τ u ) α + } = π − ( E s ) , and set α u := (1 − τ u ) α − + (1 + τ u ) α + . Consider a transverse plane field η and define || . || (cid:12)(cid:12) π − ( E u ) such that for a unit e u orienting π − ( E u ) ∩ η , we have α u ( e u ) = 1. Note that we can rewrite α t := (1 − t ) α − + (1 + t ) α + as α t = α u − ( t − τ u ) β s ;where β s = − α + + α − is a C β s ( π − ( E s )) > − α − , α + ), define || . || (cid:12)(cid:12) π − ( E s ) and let e s be the unitvector orienting π − ( E s ) ∩ η . Note that ( e s , e u , X ) is an oriented basis for M (see Figure 2). Usingthe vector bundle isomorphism T M/ (cid:104) X (cid:105) (cid:39) η , we can extend such norm to a Riemannian metric ˆ g on T M/ (cid:104) X (cid:105) with ˆ g ( E s , E u ) = 0.Let α Tu and α Ts be the C -approximations of α u and α s , which are C , in the same fashion asabove and define τ Tu , such thatker α Tu = ker { (1 − τ Tu ) α − + (1 + τ Tu ) α + } . Note that τ Tu is C and we can rewrite α t = f Tu α Tu − ( t − τ Tu ) β s for continuous function f Tu = α t ( e u ) (cid:12)(cid:12) t = τ Tu (but f Tu α Tu is C , since every other term in the aboveequation is C ). 16bserve that lim T → + ∞ τ Tu = τ u , since lim T → + ∞ ker α Tu = ker α u (Lemma 4.5). Plug in e u into α t at t = τ Tu to get f Tu α Tu ( e u ) = 1 + ( τ u − τ Tu ) β s ( e u )and in particular, lim T → + ∞ f Tu α Tu ( e u ) = 1 . (3)Similarly, plug in e s into α t at t = τ Tu to get α Tu ( e s ) = ( τ u − τ Tu ) β s ( e s ) f Tu . Compute X · ( α Tu ( e s )) =[ X · ( τ u − τ Tu ) β s ( e s ) + ( τ u − τ Tu ) X · ( β s ( e s ))] f Tu α Tu ( e u )( f Tu α Tu ( e u )) − [ X · ( τ u − τ Tu ) β s ( e u ) + ( τ u − τ Tu ) X · ( β s ( e u ))]( τ u − τ Tu ) β s ( e s ) α Tu ( e u )( f Tu α Tu ( e u )) = A ( x )( τ u − τ Tu ) + B ( x ) X · ( τ u − τ Tu )( f Tu α Tu ( e u )) for bounded functions A and B = f Tu β s ( e s ) α Tu ( e u ) + β s ( e u ) β s ( e s ) α Tu ( e u )( τ u − τ Tu ).Since lim T → + ∞ X · ( α Tu ( e s )) = 0 and B is non-zero for large T , we havelim T → + ∞ X · ( τ u − τ Tu ) = 0 , implying lim T → + ∞ X · [ f Tu α Tu ( e u )] = lim T → + ∞ { X · ( τ u − τ Tu ) β s ( e u ) + ( τ u − τ Tu ) X · ( β s ( e u )) } = 0 . (4)Also note thatlim T → + ∞ X · [ f Tu α Tu ( e s )] = lim T → + ∞ { X · ( τ u − τ Tu ) β s ( e s ) + ( τ u − τ Tu ) X · ( β s ( e s )) } = 0 . (5)Now if α := { α t } t ∈ [ − , and ω := dα , compute ω = d ( f Tu α Tu ) − [ dt − dτ Tu ] β s − ( t − τ Tu ) dβ s ;0 < ω ∧ ω (cid:12)(cid:12) t = τ Tu = dt ∧ {− β s ∧ d ( f Tu α Tu ) + ( t − τ Tu ) β s ∧ dβ s } (cid:12)(cid:12) t = τ Tu = dt ∧ {− β s ∧ d ( f Tu α Tu ) } . Compute [ β s ∧ d ( f Tu α Tu )]( e s , e u , X ) == β s ( e s )[ − X. ( f Tu α Tu ( e u )) − f Tu α Tu ( −L X e u )] − β s ( e u )[ X. ( f Tu α Tu ( e s )) − f Tu α Tu ( −L X e s )]Now by (3), (4), (5) and Proposition 3.13:0 < ω ∧ ω (cid:12)(cid:12) t = τ u = lim T → + ∞ ω ∧ ω (cid:12)(cid:12) t = τ Tu =17 lim T → + ∞ − dt ∧ β s ∧ d ( f Tu α Tu ) = β s ( e s ) r u dt ∧ ˆ e s ∧ ˆ e u ∧ ˆ X. Therefore, r u > r s <
0. This shows hyperbolicity of the splitting
T M/ (cid:104) X (cid:105) = E s ⊕ E u . By Proposition 1.1 of [14], this is equivalent to the flow being Anosov. Remark 4.9.
By Theorem 4.1, if ( ξ − , ξ + ) is a supporting bi-contact structure for an Anosovflow, then ξ − and ξ + are tight (Theorem 2.7), strongly symplectically fillable [15][21] and containno Giroux torsion[27]. Furthermore, although in general universal tightness is not achieved fromsymplectic fillability, since any lift of an Anosov flow to any cover, is also Anosov, ξ − and ξ + areuniversally tight in this case. Remark 4.10.
We note that Theorem 4.1 provides new geometric tools for understanding theperiodic orbits of Anosov flows, in particular regarding the knot theory of such periodic orbits,which there are many unanswered questions about [7]. More precisely, if γ is a periodic orbit ofan Anosov flow with supporting bi-contact structure ( ξ − , ξ + ) , then γ is a Legendrian knot for both ξ − and ξ + . Furthermore, γ × I is an exact Lagrangian in both Liouville pairs, constructed on M × I . These are standard and well-studied objects in contact and symplectic topology and now,those methods can be transferred to the study of such periodic orbits. The following theorem shows that the construction of Liouville pairs in Theorem 4.1 does notdepend on the choice of supporting bi-contact structure.
Theorem 4.11.
Let φ t be a C Anosov flow on the 3-manifold M , generated by the vector field X and ( ξ − , ξ + ) any supporting bi-contact structure for X . Then, there exist negative and positivecontact forms, α − and α + respectively, such that ker α − = ξ − , ker α + = ξ + , and ( α − , α + ) and ( − α − , α + ) are Liouville pairs.Proof. Let τ u and τ s ∈ [ − ,
1] be such that(1 − τ u ) ξ − + (1 + τ u ) ξ + = E ss ⊕ X and (1 − τ s ) ξ − + (1 + τ s ) ξ + = E uu ⊕ X, respectively.Furthermore, let e u ∈ E uu and e s ∈ E ss be the unit vector fields, corresponding to a Riemannianmetric for which r s < < r u , and respecting the orientation of E ss and E uu as in Figure 2. Define α u such that α u ( e u ) = 1 and α u ( E ss ⊕ X ) = 0. Similarly, define α s . Now, let α pre + and α pre − be thecontact forms for ξ + and ξ − , respectively, such that(1 − τ u ) α pre − + (1 + τ u ) α pre + = α u and (1 − τ s ) α pre − + (1 + τ s ) α pre + = α s . As in proof of Theorem 4.1, we can approximate α pre − and α pre + with C α − and α + , byjust scaling by suitable functions, since the plane fields are already C , such thatker α − = ker α pre − = ξ − , ker α + = ker α pre + = ξ + , and, ( α − , α + ) and ( − α − , α + ) are Liouville pairs.18 Uniqueness Of The Underlying (Bi-)Contact Structures
In this section, we want to establish various uniqueness theorems, about the (bi)-contact structuresunderlying a given C (projectively) Anosov flow. Let X be the C vector field generating suchflow and ξ be any oriented plane field such that X ⊂ ξ . In particular, we want to establish theuniqueness, up to bi-contact homotopy (see Definition 7.1), of the supporting bi-contact structure,as well as explore the conditions under which, we can retrieve the information of such bi-contactstructure, from only one of the contact structures. First, we need a definition. Definition 5.1.
We call a vector v ∈ T p M dynamically positive (negative) , if the plane (cid:104) v (cid:105) ⊕ (cid:104) X (cid:105) can be extended to a positive (negative) contact structure ξ , such that for some ξ − ( ξ + ), thereexists a supporting bi-contact structure ( ξ − , ξ ) ( ( ξ, ξ + ) ) for X . We call a vector field v dynamicallypositive (negative) on the set U ⊂ M , if it is dynamically positive (negative) at every p ∈ U .Finally, we call a plane field ξ dynamically positive (negative) on the set U ⊂ M , if ξ = (cid:104) v (cid:105) ⊕ (cid:104) X (cid:105) for some dynamically positive (negative) vector field v on U . This is basically a mathematical way of saying that a vector (or vector field or a plane field) isdynamically positive (or negative) at a point, if it lies in the interior of the first or third region (thesecond or forth region) of Figure 1 (b). Note that if ξ + is a positive contact structure coming froma supporting bi-contact structure ( ξ − , ξ + ), by Remark 3.10, ξ + is dynamically positive everywhere.But this is not true in general. That is, a general supporting positive contact structure can bedynamically negative on a subset of the manifold. However, as we will shortly discuss, the behaviorof the contact structure can be easily understood in such regions.In particular, note that if ( ξ − , ξ + ) is a supporting bi-contact structure for X , then ξ + ( ξ − ) isdynamically positive (negative) on M .Next, we see that when a supporting positive (negative) contact structure is dynamically positive(negative) everywhere on M , it is in fact isotopic, through supporting positive (negative) contactstructures, to a positive (negative) contact structure, coming from any given supporting bi-contactstructure. In particular, such contact structure is part of a supporting bi-contact structure. Lemma 5.2.
Let ( ξ − , ξ + ) be a supporting bi-contact structure for the projectively Anosov flow,generated by X , and ξ any supporting positive contact structure, which is dynamically positiveeverywhere. Then, ξ is isotopic to ξ + , through supporting positive contact structures, which aredynamically positive everywhere.Proof. It suffices to show that linear interpolation of ξ and ξ + is through positive contact structuresand Gray’s theorem guarantees the existence of isotopy. For simplicity, we assume π − ( E s ) and π − ( E u ) are C plane fields. Otherwise, we can use the approximations used in the proof ofTheorem 4.1 and the fact that both projective Anosovity and contactness are open conditions.Choose C α s and α u such that ker α s = π − ( E u ), ker α u = π − ( E s ) and ξ + = ker α + ,where α + := α u − α s . Then, there exists C function f , such that ξ = ker f α u − α s . Letting α (cid:48) + := fα u − α S , we show that for all t ∈ [0 , α t := (1 − t ) α + + tα (cid:48) + − ξ + ξ (cid:48)− ξ (cid:48) + E u E s Figure 3: Uniqueness of the supporting bi-contact structureis a positive contact structure.Choose some transverse plane field η and assume e s ∈ π − ( E s ) ∩ η and e u ∈ π − ( E u ) ∩ η arethe vector fields defined by α s ( e s ) = α u ( e u ) = 1, and r s and r u are the corresponding growth ratesof stable and unstable directions, respectively, i.e. −L X e s = r s e s − q ηs X and − L X e u = r u e u − q ηs X, for some real functions q ηs , q ηu (see Proposition 3.13).We can easily compute (as in proof of Theorem 4.1 and using Proposition 3.13):4( α ∧ dα )( e s , e u , X ) = (cid:0) α u ∧ dα u − α u ∧ dα s − α s ∧ dα u + α s ∧ dα s (cid:1) ( e s , e u , X )= − α u ( e u ) α s ([ e s , X ]) + α s ( e s ) α u ([ e u , X ]) = r u − r s > α ∧ dα )( e s , e u , X ) = (cid:0) f α u ∧ d ( f α u ) − f α u ∧ dα s − α s ∧ d ( f α u ) + α s ∧ dα s (cid:1) ( e s , e u , X )= − f α u ( e u ) α s ([ e s , X ]) + α s ( e s ) [ X · ( f α u ( e u )) + f α u ([ e u , X ])] = f r u − f r s + X · f > α ∧ dα )( e s , e u , X ) = (cid:0) α u ∧ d ( f α u ) − α u ∧ dα s − α s ∧ d ( f α u ) + α s ∧ dα s (cid:1) ( e s , e u , X )= − α u ( e u ) α s ([ e s , X ]) + α s ( e s ) [ X · ( f α u ( e u )) + f α u ([ e u , X ])] = f r u − r s + X · f ;4( α ∧ dα )( e s , e u , X ) = (cid:0) f α u ∧ dα u − f α u ∧ dα s − α s ∧ dα u + α s ∧ dα s (cid:1) ( e s , e u , X )= − f α u ( e u ) α s ([ e s , X ]) + α s ( e s ) α u ([ e u , X ]) = r u − f r s . It yields( α t ∧ dα t )( e s , e u , X ) = t ( f r u − f r s + X · f )+(1 − t ) ( r u − r s )+ t (1 − t )( r u − r s + f r u − f r s + X · f ) > , completing the proof.This, in particular, implies that the supporting bi-contact structure for any projectively Anosovflow is unique, up to homotopy through supporting bi-contact structures. Theorem 5.3. If ( ξ − , ξ + ) and ( ξ (cid:48)− , ξ (cid:48) + ) are two supporting bi-contact structures for a projectivelyAnosov flow, generated by X , then they are homotopic through supporting bi-contact structures.Proof. By Remark 3.10, ξ (cid:48) + and ξ (cid:48)− are dynamically positive and negative everywhere, respectively.The proof of Lemma 5.2 finishes the proof (See Figure 3).20t is important to understand how a positive contact structure ξ with projectively Anosov vectorfield X ⊂ ξ behaves in a region, where it is dynamically negative (similarly, we can describe thebehavior of a negative contact structure in a region, where it is dynamically positive). Consider atransverse plane field η , which is C in X -direction, and using any Riemannian metric as describedabove, define the function θ ξ : M → [0 , π ) , which measures the angle between ξ ∩ η and the bi-sector of η ∩ π − ( E s ) and η ∩ π − ( E u ) in thepositive region. Note that this function is continuous and differentiable with respect to X , where ξ is dynamically negative, ξ = π − ( E s ) or ξ = π − ( E u ). Remark 3.10 guarantees that at such points X · θ ξ < , since at those points, the flow rotates ξ clockwise in those regions (see Figure 1 (b)) and by Frobeniustheorem, ξ needs rotate faster in a clockwise fashion, to stay a positive contact structure.Now consider the family of plane fields η θ := (cid:104) X (cid:105) ⊕ l θ , for θ ∈ I − := [ π , π ] ∪ [ π , π ], where l θ ⊂ η is the oriented line field which has angle θ withthe dynamically positive bi-sector of η ∩ π − ( E s ) and η ∩ π − ( E u ). Note that such l θ is eitherdynamically negative, or the same as E s or E u (ignoring the orientation). After a generic smoothperturbation of ξ , we can assume the set Σ θ := { x ∈ M s.t. ξ = η θ } is a differentiable manifold,which is transverse to X , since X · θ ξ < the implicit function theorem ). Hence, suchsolution set is a union of tori, since the splitting T M/ (cid:104) x (cid:105) (cid:39) E s ⊕ E u would trivialize the tangentspace of such surface. Therefore, if N ⊂ M is the set on which ξ is dynamically negative, then¯ N = Σ := (cid:91) θ ∈ I − Σ θ (cid:39) (cid:91) ≤ i ≤ k T i × [0 , , for some integer k , where T i s are tori and for each 1 ≤ i ≤ k and τ ∈ [0 , X is transverse to T i × { τ } .From the above observations and what we know about Anosov flows, we can derive a host ofuniqueness theorems about ξ . Lemma 5.4.
Using the above notations, let X be an Anosov flows and N ⊂ M , the subset of M on which the positive contact structure ξ is dynamically negative.a) ¯ N (cid:39) (cid:83) ≤ i ≤ k T i × [0 , , where T i s are incompressible tori;b) ∂ ¯ N = { x ∈ M s.t. ξ = π − ( E s ) } ∪ { x ∈ M s.t. ξ = π − ( E u ) } ;c) If T , T , ..., T j of part (a) are parallel through transverse tori, then there exists a map ( S × S × [0 , ( j − π ] with coordinates ( s, t, θ ) , ker { cos θ dt + sin θ ds } ) → ( M, ξ ) , which is a contact embedding on ( S × S × [0 , ( j − π )) .d) If we assume X to be only projectively Anosov (not necessarily Anosov), we can conclude allthe above, except T i might not be incompressible.Proof. Part (a) and (b) follow from the above discussion and the fact that any surface which istransverse to an Anosov flow is an incompressible torus [12][24][40]. For Part (c), notice that if weconsider two immediate tori, we have a half-twist of the flow (a Giroux π -torsion) in between (see21emark 2.10). More precisely, we can reparametrize the angle θ of the above discussion, by theflowlines, when in the region between any two adjacent tori, where the flow is dynamically positive(and where the flow is dynamically negative, we automatically have X · θ ξ < (cid:0) [0 , π ] × S × S with coordinates ( t, φ , φ ) , ker { cos t dφ + sint dφ (cid:9) ) → ( M, ξ ) . Theorem 5.5. If M is atoroidal and ( ξ − , ξ + ) a supporting bi-contact structure for the Anosovvector field X on M , then for any supporting positive contact structure ξ , ξ is isotopic to ξ + ,through supporting contact structures.Proof. By Lemma 5.4 ξ is dynamically positive everywhere, and Lemma 5.2 finishes the proof.An Anosov flow is called R -covered , if the lift of its stable (or unstable) foliations to the universalcover is the product foliation of R by planes. This is an important class of Anosov flows and isstudied in depth, in the works of Fenley, Bartbot, Barthelme, etc. In particular, it is shown [5][23]that there is no embedded surface transverse to such flows. Hence, Theorem 5.6.
Let X be an R -covered Anosov vector field, supported by the bi-contact structure ( ξ − , ξ + ) on M , and let ξ be any supporting positive contact structure. Then, ξ is isotopic to ξ + ,through supporting contact structures. In the case of M being a torus bundle, the underlying contact structures can be characterizedby having the minimum torsion (see Remark 2.10). Although, similar phenomena can be observedin the case of projective Anosov flows, we state the theorem for Anosov flows, for which the relationof torsion and symplectic fillability is established in [27][38]. The proof relies on the classificationof contact structures on torus bundles and T × I by Ko Honda and one should consult [32] and[33] for more details and precise definitions. Theorem 5.7.
Let X be the suspension of an Anosov diffeomorphism of torus, supported by thebi-contact structure ( ξ − , ξ + ) , and ξ a positive supporting contact structure. Then, ξ is isotopicthrough supporting bi-contact structures to ξ + , if and only if, ξ is strongly symplecitcally fillable.Proof. If ξ is dynamically positive everywhere, Lemma 5.2 yields the isotopy. Otherwise, for anyincompressible torus T i in Lemma 5.4, the flow is a suspension flow for an appropriate Anosovdiffeomorphism of T i . Notice that there is at least two of such T i , since ξ is coorientable. The ideais that, in this case, ξ rotates at least 2 π more than ξ + , as we move in S -direction (see Lemma 5.4)and since ξ + rotates some itself, that means that ξ rotates more than 2 π . Therefore, ξ containsGiroux torsion and is not symplectically fillable.Let ˜ M := M \ T (cid:39) T × I , where we have compactified M \ T , by gluing two copies of T alongthe boundary. i.e. T and T , such that ∂ ˜ M = − T (cid:116) T (abusing notation, we call the inducedcontact structures, ξ + and ξ ). After a choice of basis for T , let s i + ( s i ), i = 1 ,
2, be the slope ofthe characteristic foliation of ξ + ( ξ ) on T i , respectively. That is the foliation of T i by T T i ∩ ξ + ( T T i ∩ ξ ).Note that since ξ + is universally tight, by [33], ξ + has nonnegative twisting as it goes from T to T . Furthermore, since ξ + does not contain Giroux torsion, such twisting is less than 2 π . Weclaim that s (cid:54) = s and s (cid:54) = s . That is because an Anosov diffeomorphism of torus preservesexactly two slopes of the torus and those are the intersections of π − ( E s ) and π − ( E u ) with theboundary. 22ow by Lemma 5.4 c), there exist at least a π -twisting between T and T , as well as between T and T . That is a total of at least 2 π -twisting. i.e. a contact embedding (cid:0) [0 , π ] × S × S with coordinates ( t, φ , φ ) , ker { cos t dφ + sint dφ } (cid:1) → ( ˜ M , ξ ) , with { } × S × S → T . Since Im ( { } × S × S ) has the same slope s as T (after a 2 π -twist),this implies Im ( { } × S × S ) ∩ T = ∅ and therefore, we will achieve an embedding (cid:0) [0 , π ] × S × S with coordinates ( t, φ , φ ) , ker { cos t dφ + sint dφ } (cid:1) → ( M, ξ ) , meaning that ξ contains Giroux torsion. In this section, we use ideas developed in Section 3 and the proof of Theorem 4.1 to give a char-acterization of Anosovity, based on the
Reeb flows , associated to the underlying contact structuresof a projectively Anosov flow. First, recall:
Definition 6.1.
Given a contact manifold ( M, ξ ) , for any choice of contact form α for ξ , thereexists a unique vector field R α , named the Reeb vector field with the following properties: α ( R α ) = 1 , dα ( R α , . ) = 0 . Example 6.2.
1) For the standard tight contact structure on S , described in Example 2.2 2), theReeb vector field is tangent to Hopf fibration, for an appropriate choice of contact form.2) For all the contact structures of Example 2.2 3) on T , the unit orthogonal vector field(considering the flat metric on T (cid:39) R ), is the Reeb vector field.3) The geodesic flow on the unit tangent space of a hyperbolic surface is the Reeb vector field forthe tautological 1-form (which is a contact form). In this case the Reeb vector field itself is Anosov. Theorem 6.3.
Let X be a projectively Anosov vector field on M . Then, the followings are equiv-alent:(1) X is Anosov;(2) There exists a supporting bi-contact structure ( ξ − , ξ + ) , such that ξ + admits a Reeb vectorfield, which is dynamically negative everywhere;(3) There exists a supporting bi-contact structure ( ξ − , ξ + ) , such that ξ − admits a Reeb vectorfield, which is dynamically positive everywhere.Proof. For simplicity assume π − ( E s ) and π − ( E u ) are C plane fields. The general case followsfrom the approximations described in the proof of Theorem 4.1 and the fact that Anosovity, as wellas being dynamically positive (negative) everywhere, are open conditions.Assuming (1), we now show (2).Choose a transverse plane field η and let e s ∈ π − ( E s ) ∩ η and e u ∈ π − ( E u ) ∩ η be the unitvector fields with respect to the Riemannian metric satisfying r s < < r u , and α s defined by α s ( e s ) = 1 and α s ( π − ( E u )) = 0 (see proof of Theorem 4.1 for notation). Similarly, define α u .Define α + := ( α u − α s ). Note that ( ξ − , ξ + := ker α + ) is a supporting bi-contact structure, for23n appropriate choice of ξ − . The span of the Reeb vector field, R α + , is determined by the twoequations dα + ( X, R α + ) = 0 = dα + ( e + , R α + ) , where e + ∈ ξ + is a vector field such that that (cid:104) X (cid:105) ⊕ (cid:104) e + (cid:105) = ξ + .Consider the vector v := − r s e u − r u e s and note that since r s < < r u , such vector is dynamicallynegative. Compute dα + ( X, v ) = − r s dα + ( X, e u ) − r u dα + ( X, e s ) = − r s r u + r s r u = 0 . This implies R α + ⊂ (cid:104) X, v (cid:105) and therefore, R α + is dynamically negative everywhere.Now assume (2) and we establish (1).Let α + be such contact form for ξ + . Define α u and α s such that α + = ( α u − α s ) and α s ( π − ( E u )) = α u ( π − ( E s )) = 0. Finally, choose a transverse plane field η and define the Rie-mannian metric such that for unit vectors e s ∈ π − ( E s ) ∩ η and e u ∈ π − ( E u ) ∩ η , we have α s ( e s ) = α u ( e u ) = 1. By the above computation, we observe R α + ⊂ (cid:104) X, − r s e u − r u e s (cid:105) . Since suchvector is dynamically negative, this implies r s < < r u , and therefore, X is Anosov.Equivalence of (1) and (3) is similar. In Theorem 4.1, we proved that the Anosovity of a flow is equivalent to a host of contact andsymplectic geometric conditions. This bridge naturally creates a hierarchy of geometric conditionson the flow and therefore, a new filtration of Anosov dynamics, starting with projectively Anosovflows and ending with Anosov flows. It is of general interest to understand which layer of geo-metric conditions is responsible for properties of Anosov flows and introduces new geometric andtopological tools to study questions in Anosov dynamics. In this section, we want to establish suchhierarchy, make some remarks and formalize a platform for such study.In Theorem 4.1, we observed that underlying any Anosov flow is a bi-contact structure, corre-sponding to the projective Anosovity of the flow. In order to reduce the questions about Anosovdynamics to contact topological questions, we first need to understand the dependence of Anosovityon the geometry of the supporting bi-contact structure. First, we define two notions of equivalencefor bi-contact structure, which can describe deformation of a projectively Anosov flow.
Definition 7.1.
We call two bi-contact structures ( ξ − , ξ + ) and ( ξ (cid:48)− , ξ (cid:48) + ) bi-contact homotopic , ifthere exists a homotopy of bi-contact structures ( ξ t − , ξ t + ) , t ∈ [0 , with ( ξ − , ξ ) = ( ξ − , ξ + ) and ( ξ − , ξ ) = ( ξ (cid:48)− , ξ (cid:48) + ) . We call the two bi-contact structures isotopic , if such homotopy is induced byan isotopy of the underlying manifold. Note that bi-contact homotopy of ( ξ − , ξ + ) and ( ξ (cid:48)− , ξ (cid:48) + ) is equivalent to the supported projec-tively Anosov flows to be homotopic through projectively Anosov flows. Also in this case, by Gray’stheorem, ξ − and ξ (cid:48)− , as well as ξ + and ξ (cid:48) + , are isotopic, but not necessary through the same isotopy.We believe that the notion of bi-contact isotopy is too rigid for the study of Anosov dynamics,while bi-contact homotopy is more natural. Because firstly, given a fixed Anosov flow and two24upporting bi-contact structures, a priori, the two bi-contact structures are not isotopic, while theyare bi-contact homotopic through supporting bi-contact structures, by Theorem 5.3. Secondly,the isotopy class of bi-contact structures in unstable even under C -perturbation of the supportedvector field, since such isotopy forces the eigenvalues of the Poincare return map of any periodicorbit to be fixed, while such eigenvalues can vary with deformations.The dependence of Anosovity on bi-contact homotopy is yet to be understood. Question 7.2.
Let ( ξ − , ξ + ) be a bi-contact structure, supporting an Anosov flow, and ( ξ (cid:48)− , ξ (cid:48) + ) another bi-contact structure which is bi-contact homotopic to ( ξ (cid:48)− , ξ (cid:48) + ) . Are projectively Anosovflows, supported by ( ξ (cid:48)− , ξ (cid:48) + ) Anosov?
While an affirmative answer to the above question might be too optimistic, confirming thefollowing more modest conjecture can still reduce many problems in Anosov dynamics, regardingthe orbit structures and periodic orbits, to contact topological problems.
Conjecture 7.3.
Two Anosov flows which are supported by bi-contact homotopic bi-contact struc-tures are orbit equivalent. That is, there exists a homeomorphism of the manifold sending theorbits of one to the other. Equivalently, two Anosov flows which are homotopic through projectivelyAnosov flows are orbit equivalent.
A weaker notion than bi-contact homotopy, is when given two bi-contact structures, the positivecontact structures, as well as the negative contact structures, are isotopic. But the transversality ofthe two might be violated during the homotopy. In other words, can a pair of positive and negativecontact structures be transverse in two distinct ways?
Question 7.4.
Let ( ξ − , ξ + ) and ( ξ (cid:48)− , ξ (cid:48) + ) be two bi-contact structures, such that ξ − and ξ (cid:48)− , as wellas ξ + and ξ (cid:48) + , are isotopic. Are ( ξ − , ξ + ) and ( ξ (cid:48)− , ξ (cid:48) + ) bi-contact homotopic? After questions regarding the relation of Anosovity and the geometry of the supporting bi-contact structure, we can ask about the relation to the topology of bi-contact structures. Moreprecisely, Anosovity implies rigid contact and symplectic topological properties of the underlyingcontact structures (see Theorem 4.1 and Remark 4.9). It is natural to ask about the degree towhich these topological properties are responsible for the dynamical properties of Anosov flows.
Definition 7.5.
We call a bi-contact ( ξ − , ξ + ) structure tight , if both ξ − and ξ + are tight, and wecall a projectively Anosov flow tight , if it is supported by a tight bi-contact structure. Here, we conjecture that tightness of one of the contact structures implies the above condition.
Conjecture 7.6. If ( ξ − , ξ + ) is a bi-contact structure, then ξ − and ξ + are either both tight, or bothovertwisted. In [41], Mitsumatsu introduces a criteria of making a pair positive and negative contact struc-tures transverse, which yields tight bi-contact structures on T and nil-manifolds (note that theseprojectively Anosov flows are not Anosov [43]). Here, we put down the explicit examples on T . Example 7.7.
After an isotopy, and for integers m, n > , we consider the positive and negativetight contact structures of Example 2.2 3) on T , ξ n = ker dz + (cid:15) { cos 2 πnzdx − sin 2 πnzdy } and ξ − m = ker dz + (cid:15) (cid:48) { cos 2 πmzdx + sin 2 πmzdy } , respectively. It is easy to observe that if (cid:15) (cid:54) = (cid:15) (cid:48) , then ξ n and ξ − m are transverse everywhere, and therefore, their intersection contains tight projectivelyAnosov vector fields. efinition 7.8. We call a bi-contact structure ( ξ − , ξ + ) on M weakly, strongly or exactly sym-plectically bi-fillable , if there exists ( X, ω ) , which is a weak, strong or exact symplectically fillingfor ( M, ξ + ) (cid:116) ( − M, ξ − ) , respectively, where − M is M with reversed orientation. We call a projec-tively Anosov flow supported by such bi-contact structure, weakly, strongly or exactly symplecticallybi-fillable , respectively. Furthermore, we call the symplectic bi-filling trivial , if X (cid:39) M × [0 , . Note that any Anosov flow is exactly symplectically bi-fillable. Furthermore, any exactlybi-fillable projectively Anosov flow is strongly bi-fillable and any strongly bi-fillable projectivelyAnosov flow is weakly bi- fillable.Using the idea of Example 2.9 2), we can show that these projectively Anosov flows are in fact,weakly bi-fillable.
Theorem 7.9.
The tight projectively Anosov flows of Example 7.7 are trivially weakly symplecticallybi-fillable.Proof.
Let X = T × A , where A is an annulus. Consider the coordinates ( x, y ) for T and let z be the angular coordinate of A , near its boundary. If ω and ω are some area forms on T and A ,respectively, then ω = ω ⊕ ω will be a symplectic form on X , such that ω | ker dz >
0. Choosingsmall enough (cid:15), (cid:15) (cid:48) > ξ n and ξ − m would be arbitrary close to ker dz and therefore, ω | ξ n , ω | ξ − m >
0, implying that ( ξ − m , ξ n ) is weakly symplectically bi-fillable, for any pair of integers m, n > ξ n and ξ − m to be strongly symplectically bi-fillable, which is notthe case [19]). Corollary 7.10.
There are (trivially) weakly symplectically bi-fillable projectively Anosov flows,which are not strongly symplectically bi-fillable.
The properness of other inclusions in the described hierarchy of projectively Anosov flows re-mains an open problem.
Question 7.11.
Are there tight projectively Anosov flows, which are not weakly symplectically bi-fillable? Are there strongly bi-fillable projectively Anosov flows, which are not exactly bi-fillable?Are there exactly bi-fillable projectively Anosov flows, which are not Anosov?
Remark 7.12.
Here, we remark that our filtration of contact and symplectic conditions on aprojectively Anosov flow is what we found more natural and can be refined and modified in otherways and using other conditions, for instance on the topology of the symplectic fillings, etc. Inparticular, we note that in our definition of symplectic bi-fillings for a projectively Anosov flow,supported by a bi-contact structure ( ξ − , ξ + ) , we did not consider bi-fillability for both ( ξ − , ξ + ) and ( − ξ − , ξ + ) , a condition which is satisfied for Anosov flows. Note that symplectic fillability, fora contact manifold ( M, ξ ) with connected boundary, does not depend on the coorientation of thecontact structure, since if ( X, ω ) is a symplectic filling for such contact manifold, then ( X, − ω ) would be a symplectic filling for ( M, − ξ ) . But when we have disconnected boundary, like in the caseof bi-contact structures, fillability properties might change if we flip the orientation of only one ofthe contact structures. eferences [1] Anosov, Dmitry Victorovich. Geodesic flows on closed Riemannian manifolds of negative cur-vature.
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Surena Hozoori,
Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332
E-mail address : [email protected]@gatech.edu