Totally non-symplectic Anosov actions on tori and nilmanifolds
aa r X i v : . [ m a t h . D S ] M a r TOTALLY NON-SYMPLECTIC ANOSOV ACTIONS ON TORIAND NILMANIFOLDS
DAVID FISHER, BORIS KALININ, RALF SPATZIER
Abstract.
We show that sufficiently irreducible totally non-symplectic Anosovactions of higher rank abelian groups on tori and nilmanifolds are C ∞ -conjugateto actions by affine automorphisms. Introduction
Hyperbolic actions of abelian groups of rank at least 2 exhibit many surprisingrigidity properties. Case in point is the local smooth rigidity of actions by automor-phisms of tori and nilmanifolds and other algebraically defined actions. This meansthat perturbations of an action that are C -close for a finite set of generators are C ∞ -conjugate to the original action. It was established for algebraic actions withsemisimple linear part by Katok and Spatzier in [19] and for some non-semisimpleactions on tori by Einsiedler and T. Fisher [4]. The higher rank situation is entirelydifferent from the case of single Anosov diffeomorphisms and flows for which it isalways easy to construct C -small perturbations which are not even C -conjugate.Local smooth rigidity of algebraic actions gives strong support to the followingconjecture by Katok and Spatzier. Classification Conjecture: All “irreducible” Anosov Z k and R k -actions for k ≥ on any compact manifold are C ∞ -conjugate to algebraic actions. Kalinin and Spatzier proved this conjecture for the special class of Cartan actionsof abelian groups of rank at least 3 under some other more technical hypotheses [16].Here we call an action
Cartan if maximal intersections of stable manifolds of variouselements, called coarse Lyapunov foliations , are one-dimensional and, together withthe orbit, span the space. Kalinin and Sadovskaya have results for more generalAnosov actions of rank at least 2 where the condition on dimension 1 is replaced byeither uniform quasi-conformality or a pinching condition [14, 15]. The basic idea ofthe proofs in all of these results is to build smooth structures on various foliations
Date : November 21, 2018.First author supported in part by NSF grant DMS-0643546. Second author supported in partby NSF grant DMS-0701292. Third author supported in part by NSF grants DMS-0906085 andDMS-0604857. and then combine them. Unfortunately, this only works under strong assumptionson the action.The general case of the conjecture remains out of reach. Thus it is natural torestrict attention to actions on tori and nilmanifolds where one usually refers to theconjecture as global rigidity . For these spaces, the classical results of Franks andManning [6, 22] offer a different approach. Their work implies that any action α ofan abelian group with at least one Anosov element on a torus or a nilmanifold isalways C -conjugate to an action by affine Anosov automorphisms by some H¨olderconjugacy φ . Now to prove global rigidity it suffices to show smoothness of theconjugacy φ . We call the latter action the linearization of α and refer to Section 2 fora precise definition. On the torus the linearization is essentially given by the inducedaction on homology. Note that in the nilmanifold case, the term “linearization” isa bit of a misnomer as the action by automorphisms is not really linear.The idea that a C conjugacy can be used to get C ∞ -rigidity appears alreadyin Hurder’s work on deformation rigidity of lattice actions on tori [10] and laterin Katok-Lewis [18] for both their local and global rigidity theorems for Cartanactions on tori. It also formed the basis of the argument for local rigidity in Katok-Spatzier [19]. In the different context of local rigidity of algebraic actions of latticesin higher rank groups, work of Katok and Spatzier and later Fisher, Margulis andQian [5, 19, 23] also involves finding a C conjugacy that is improved to C ∞ usingthe presence of higher rank abelian subgroups in the acting group. Rodriguez Hertzestablished global rigidity for Z k actions on tori with at least one Anosov elementwhose linearization has coarse Lyapunov foliations of dimensions one or two andeither has maximal rank or satisfies additional bunching assumptions [27]. To datehowever, all results require that the derivatives of either the action or its linearizationalong the coarse Lyapunov foliations satisfy a pinching assumption. This means thatthe ratio of maximal over minimal contraction is controlled, e.g. less than 2. In thispaper, we overcome this problem for the first time by a combination of the use ofnon-stationary normal forms and holonomy arguments. Beyond achieving a superiorresult, the use of the two tools is also completely novel. We use limits of holonomymaps to define homogeneous structures on certain foliations. This has never beendone before. Furthermore, we make use of measurable normal forms for the firsttime in the context of global rigidity of actions. Previously measurable normal formshave only been used to study invariant measures.Continuous normal forms were already introduced for the proof of local rigidity in[19]. In essence they give coordinate charts in which the derivatives of the map alongcontracting foliations take values in a finite dimensional Lie group. Moreover, thedependence of the coordinates on the base point is continuous in the C ∞ -topology.Existence of continuous normal forms is guaranteed if the derivatives of the mapsunder consideration satisfy a spectral gap condition along the given contracting fo-liation. While such spectral gaps are automatic for C -perturbations of algebraic OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 3 systems and also for one dimensional foliations, they fail to hold in general. In par-ticular we cannot assume such spectral gaps for the proof of global rigidity. Instead,we use a measurable version of the non-stationary normal forms theory where the“measurable” spectral gap condition is always satisfied by Oseledec’ MultiplicativeErgodic Theorem.Let us next summarize some elements from the structure theory of higher rankabelian actions, see Section 2 for more details. They preserve a probability measureof full support. One can find a common Lyapunov splitting of the tangent bundle
T M = ⊕ i E i which refines the Lyapunov splittings of each individual element. More-over, if v ∈ E i , the Lyapunov exponent of v defines a linear functional, the Lyapunovfunctional , on the acting Z k which we think of as a linear functional on the ambient R k . For actions by affine automorphisms the Lyapunov exponents are nothing butthe logarithms of the absolute values of the eigenvalues of the automorphisms. A Weyl chamber is a connected component of R k minus all the hyperplane kernels ofthe Lyapunov functionals. We will need to make the assumption that every Weylchamber defined by the linearization contains an Anosov element in the non-linearaction. As we will later see that the Weyl chambers on the two sides agree, we ab-breviate this by saying that every Weyl chamber contains an Anosov element. Thisallows us to define the coarse Lyapunov foliations as the maximal intersections ofstable foliations of Anosov elements. Hence these foliations are H¨older with smoothleaves.Recall that a matrix is semisimple if it is diagonalizable over C . We call anaction by affine automorphisms of a nilmanifold semisimple if the linear part ofevery element acts by a semisimple matrix.Finally, we call a Z k -action TNS or totally non-symplectic if any two v ∈ E i and w ∈ E j belong to the stable distribution of some element a ∈ Z k . This excludes thepossibility of a bilinear form invariant under the action, hence the name.The main result of this paper proves global rigidity for totally non-symplecticactions. Theorem 1.1.
Suppose α is a C ∞ -action of Z k , k ≥ on a nilmanifold N/ Γ .Assume the linearization ρ of α is semisimple and TNS and there is an Anosovelement in each Weyl chamber of α . Then α is C ∞ -conjugate to ρ . As discussed above, this theorem is the first that does not require pinching con-ditions. Moreover, it also yields the first global rigidity result for Anosov actionson nilmanifolds which are not tori. Indeed, in all earlier results the pinching condi-tion, together with various additional assumptions such as integrability or absenceof certain resonances, forced the nilmanifold to be a torus.Call a linear Z k action on a torus totally reducible if every rational invariant torushas a rational invariant complement. There is a similar though more complicatednotion for nilmanifolds which we describe below in section 9. We will show that DAVID FISHER, BORIS KALININ, RALF SPATZIER total reducibility is equivalent to semisimplicity, and thus we immediately get thenext result:
Corollary 1.2.
Suppose α is a C ∞ -action of Z k , k ≥ on a nilmanifold N/ Γ .Assume the linearization ρ of α is totally reducible and TNS and there is an Anosovelement in each Weyl chamber of α . Then α is C ∞ -conjugate to ρ . To prove the corollary from the theorem, we prove that any totally reducibleaction is semisimple.Our results have some applications to global rigidity for actions of higher ranklattices. Margulis and Qian prove that any Anosov action of a higher rank lattice Γon a nilmanifold with a common fixed point for the entire group action is continu-ously conjugate to an action by affine automorphisms[23]. It is well known that suchΓ contains many abelian subgroups isomorphic to Z k , where k is the real rank of Γ,and that the Anosov Γ action restricts to an Anosov Z k action. If some Z k subgroupsatisfies the conditions of Theorem 1.1, it then follows from our results that theconjugacy is smooth, and therefore that the full Γ action is smoothly conjugate toan action by affine automorphisms.Let us briefly indicate the main elements in the proof of Theorem 1.1. As discussedabove we show that the topological conjugacy φ is smooth. For this, we first suspendthe Z k -action to an R k -action. Then we fix a coarse Lyapunov foliation and foralmost every leaf we construct a transitive group of smooth transformations whichis intertwined by φ with the group of translations of the corresponding leaf for thelinearization. As in other proofs of rigidity theorems e.g. in [19], we use limits ofreturn maps. Unlike earlier proofs however, we do not directly use the acting groupbut rather holonomies along transversal coarse Lyapunov foliations. First we showthat these holonomies are smooth. For this we establish existence of elements whichcontract the fixed coarse Lyapunov foliation slower than a transversal one. Thenwe show that the holonomies centralize suitable elements of R k and hence preservemeasurable non-stationary normal forms. It follows that limits of such holonomiesare still smooth and define the desired transitive group actions. Once the smoothnessof φ is established for a.e. leaf of each coarse Lyapunov foliation, the smoothness ofholonomies gives the global smoothness of φ . A more detailed outline of the proofis given in Section 3, after all relevant notions have been defined.We would like to thank K. Burns, D.Dolgopyat, F.Ledrappier Y. Pesin, J. Rauchand A. Wilkinson for a number of discussions on subjects related to this paper.2. Preliminaries
Throughout the paper, the smoothness of diffeomorphisms, actions, and manifoldsis assumed to be C ∞ , even though all definitions and some of the results can beformulated in lower regularity. OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 5
Anosov actions of Z k and R k . Let a be a diffeomorphism of a compact manifold M . We recall that a is Anosov if there exist a continuous a -invariant decomposition of the tangent bundle T M = E sa ⊕ E ua and constants K > λ > n ∈ N (1) k Da n ( v ) k ≤ Ke − λn k v k for all v ∈ E sa , k Da − n ( v ) k ≤ Ke − λn k v k for all v ∈ E ua . The distributions E sa and E ua are called the stable and unstable distributions of a .Now we consider a Z k action α on a compact manifold M via diffeomorphisms.The action is called Anosov if there is an element which acts as an Anosov dif-feomorphism. For an element a of the acting group we denote the correspondingdiffeomorphisms by α ( a ) or simply by a if the action is fixed.For a Z k action α on a manifold M , there is an associated R k action ˜ α on amanifold S given by the standard suspension construction [12]. Briefly, this is theaction of R k by left translations on ( R k × M ) / Z k . Here ( R k × M ) / Z k is the quotientof R k × M by the Z k -action on R k × M given by z ( r, p ) = ( r − z, z ( p )). We will referto ˜ α as the suspension of α . It generalizes the suspension flow of a diffeomorphism.Similarly, the manifold S is a fibration over the “time” torus T k with fiber M . Definition 2.1.
Let α be a smooth action of R k on a compact manifold M . Anelement a ∈ R k is called Anosov or normally hyperbolic for α if there exist positiveconstants λ , K and a continuous α -invariant splitting of the tangent bundle T M = E sa ⊕ E ua ⊕ T O where T O is the tangent distribution of the R k -orbits, and (1) holds for all n ∈ N .An R k action is called Anosov if some element a ∈ R k is Anosov. Note that a ∈ Z k is Anosov for α if and only if it is Anosov for ˜ α . Thus if α is an Anosov Z k actionthen ˜ α is an Anosov R k action.Both in the discrete and the continuous case it is well-known that the distri-butions E sa and E ua are H¨older continuous and tangent to the stable and unstablefoliations W sa and W ua respectively [9]. The leaves of these foliations are C ∞ injec-tively immersed Euclidean spaces. Locally, the immersions vary continuously in the C ∞ topology. In general, the distributions E s and E u are only H¨older continuoustransversally to the corresponding foliations.2.2. Lyapunov exponents and coarse Lyapunov distributions.
First we recall some basic facts from the theory of non-uniform hyperbolicity fora single diffeomorphism, see for example [2]. Then we consider Z k and R k actionsconcentrating on the continuous time case on the case, we refer to [16] and [14] formore details.Let a be a diffeomorphism of a compact manifold M preserving an ergodic proba-bility measure µ . By Oseledec’ Multiplicative Ergodic Theorem, there exist finitely DAVID FISHER, BORIS KALININ, RALF SPATZIER many numbers χ i and an invariant measurable splitting of the tangent bundle T M = L E i on a set of full measure such that the forward and backward Lya-punov exponents of v ∈ E i are χ i . This splitting is called Lyapunov decomposition .We define the stable distribution of a with respect to µ as E − a = L χ i < E i . Thesubspace E − a ( x ) is tangent µ -a.e. to the stable manifold W − a ( x ). More generally,given any θ < E θa = L χ i ≤ θ E i which is tangent µ -a.e. to the strong stable manifold W θa ( x ). W θa ( x ) is a smoothlyimmersed Euclidean space. For a sufficiently small ball B ( x ), the connected compo-nent of W θa ( x ) ∩ B ( x ), called local manifold, can be characterized by the exponentialcontraction property(2) W θ,loca ( x ) = { y ∈ B ( x ) | dist( a n x, a n y ) ≤ Ce ( θ + ε ) n ∀ n ∈ N } . The unstable distributions and manifolds are defined similarly. In general, E − a isonly measurable and depends on the measure µ . However, if a is an Anosov diffeo-morphism, or an Anosov element of an R k action, then E − a for any measure alwaysagrees with the continuous stable distribution E sa . Indeed, E sa cannot contain avector with a nontrivial component in some E j with χ j ≥ E sa ⊂ L χ i < E i . Similarly, the unstable distribution E ua ⊂ L χ i > E i . Since E sa ⊕ E ua is transverse to the orbit, both inclusions have tobe equalities.Let µ be an ergodic probability measure for an R k action α on a compact manifold M . By commutativity, the Lyapunov decompositions for individual elements of R k can be refined to a joint invariant splitting for the action. The following propositionfrom [16] describes the Multiplicative Ergodic Theorem for this case. See [14] forthe discrete time version and [12] for more details on the Multiplicative ErgodicTheorem and related notions for higher rank abelian actions. Proposition 2.2.
Let α be a smooth action of R k and let µ be an ergodic invariantmeasure. There are finitely many linear functionals χ on R k , a set of full measure P , and an α -invariant measurable splitting of the tangent bundle T M = T O ⊕ L E χ over P , where O is the orbit foliation, such that for all a ∈ R k and v ∈ E χ , theLyapunov exponent of v is χ ( a ) , i.e. lim t →±∞ t − log k (cid:0) Dα ( ta ) (cid:1) ( v ) k = χ ( a ) , where k .. k is a continuous norm on T M . The splitting L E χ is called the Lyapunov decomposition , and the linear function-als χ are called the Lyapunov exponents of α . The hyperplanes ker χ ⊂ R k are calledthe Lyapunov hyperplanes or Weyl chamber walls , and the connected components of R k − ∪ χ ker χ are called the Weyl chambers of α . The elements in the union of theLyapunov hyperplanes are called singular , and the elements in the union of the Weylchambers are called regular . We note that the corresponding notions for a Z k actionand for its suspension are directly related. In particular, the nontrivial Lyapunov OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 7 exponents are the same. In addition, for the suspension there is one identically zeroLyapunov exponent corresponding to the orbit distribution. From now on, the termLyapunov exponent will always refer to the nonzero functionals.Consider a Z k action by automorphisms of a torus or a nilmanifold M = N/ Γ.In this case, the Lyapunov decomposition is determined by the eigenspaces of theautomorphisms, and the Lyapunov exponents are the logarithms of the moduli ofthe eigenvalues. Hence they are independent of the invariant measure, and theygive uniform estimates of expansion and contraction rates. Also, every Lyapunovdistribution is smooth and integrable.In the non-algebraic case, the individual Lyapunov distributions are in generalonly measurable and depend on the given measure. This can be already seen fora single diffeomorphism, even if Anosov. However, as we observed above, the fullstable distribution E sa of an Anosov element a always agrees with L χ ( a ) < E χ on aset of full measure for any measure.For higher rank actions, coarse Lyapunov distributions play a similar role to thestable and unstable distributions for an Anosov diffeomorphism. For any Lyapunovfunctional χ the coarse Lyapunov distribution is the direct sum of all Lyapunovspaces with Lyapunov functionals positively proportional to χ : E χ = ⊕ E χ ′ , χ ′ = c χ with c > . For an algebraic action such a distribution is a finest nontrivial intersection ofthe stable distributions of certain Anosov elements of the action. For nonalgebraicactions, however, it is not a priori clear. It was shown in [16, Proposition 2.4]that, in the presence of sufficiently many Anosov elements, the coarse Lyapunovdistributions are well-defined, continuous, and tangent to foliations with smoothleaves (see Proposition 2.2 in [15] for the discrete time case). We denote the set ofall Anosov elements in Z k or R k by A . Proposition 2.3.
Let α be an Anosov action of Z k or R k and let µ be an ergodicprobability measure for α with full support. Suppose that there exists an Anosovelement in every Weyl chamber defined by µ . Then for each Lyapunov exponent χ the coarse Lyapunov distribution can be defined as E χ ( p ) = \ { a ∈A | χ ( a ) < } E sa ( p ) = M { χ ′ = c χ | c> } E χ ′ ( p ) on the set P of full measure where the Lyapunov exponents exist. Moreover, E χ isH¨older continuous, and thus it can be extended to a H¨older distribution tangent tothe foliation W χ = T { a ∈A | χ ( a ) < } W sa with uniformly C ∞ leaves. Note that ergodic measures with full support always exist if a Z k action containsa transitive Anosov element. DAVID FISHER, BORIS KALININ, RALF SPATZIER
A natural example is given by the measure µ of maximal entropy for such anelement, which is unique [17, Corollary 20.1.4] and hence is invariant under thewhole action.Since a coarse Lyapunov distribution is defined by a collection of positively pro-portional Lyapunov exponents it can be uniquely identified with the correspondingpositive (negative) set of these functionals, called the positive (negative) Lyapunovhalf-space , or with the oriented Lyapunov hyperplane that separates them.The action is called totally nonsymplectic , or TNS , if there are no nega-tively proportional Lyapunov exponents. Equivalently, any two different negativeLyapunov half-spaces have nontrivial intersection. Therefore, any pair of coarseLyapunov distributions for such an action is contracted by the elements in thisintersection.2.3. Z k and R k actions on tori and nilmanifolds. Let f be an Anosov diffeo-morphism of a torus or, more generally, a nilmanifold M = N/ Γ. By the resultsof Franks and Manning in [6, 22], f is topologically conjugate to an Anosov au-tomorphism A : M → M , i.e. there exists a homeomorphism φ : M → M suchthat A ◦ φ = φ ◦ f . The conjugacy φ is bi-H¨older, i.e. both φ and φ − are H¨oldercontinuous with some H¨older exponent γ .Now we consider an Anosov Z k action α on a nilmanifold M . Fix an Anosovelement a for α . Then we have φ which conjugates α ( a ) to an automorphism A .By [29, Corollary 1] any homeomorphism of M commuting with A is an affineautomorphism. Hence we conclude that φ conjugates α to an action ρ by affineautomorphisms. We will call ρ an algebraic action and refer to it as the linearization of α .Now we describe the preferred invariant measure for α (cf. [13, Remark 1]). Wedenote by λ the normalized Haar measure on the nilmanifold M . Note that λ isinvariant under any affine automorphism of M and is the unique measure of maximalentropy for any affine Anosov automorphism. Proposition 2.4.
The action α preserves an absolutely continuous measure µ withsmooth positive density. Moreover, µ = φ − ∗ ( λ ) and for any Anosov element a ∈ Z k , µ is the unique measure of maximal entropy for α ( a ) .Proof : Let J denote the Jacobian of α with respect to the Haar measure λ ,more precisely, for any b ∈ Z k the function J b ( x ) denotes the density of the pushforward measure α ( b ) ∗ ( λ ) with respect to λ . Since the conjugacy φ is bi-H¨older,log ( J ◦ φ − ) is a H¨older cocycle over the linearization ρ . By rigidity of H¨older cocyclesfor irreducible algebraic Z k -actions [19], this cocycle is H¨older cohomologous to aconstant one, i.e. there exists a linear functional c : Z k → R and a H¨older continuousfunction Φ on M such that for all b ∈ Z k and x ∈ M log ( J b ◦ φ − ( x )) = c ( b ) + Φ( ρ ( b ) x ) − Φ( x ) . OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 9 a constant c . Hence for the function Ψ = Φ ◦ φ we havelog ( J b ( x )) = c ( b ) + Ψ( α ( b ) x ) − Ψ( x ) . Let µ be the measure e − Ψ( x ) λ normalized by µ ( M ) = 1. It follows that the Jacobian˜ J of α with respect to µ is constant, ˜ J b ( x ) = e c ( b ) . Since α ( b ) is a diffeomorphismswe have R M ˜ J b ( x ) dµ = 1 and hence e c ( b ) = 1. Thus α preserves µ .Let a be an Anosov element of α . Then the density of its absolutely continuousmeasure µ is C [17, Theorem 19.2.7], and in fact C ∞ [21, Corollary 2.1]. Also, µ isthe equilibrium state of log ( J u ) [17, Theorem 20.4.1], where J u is the Jacobian of α ( a ) along its unstable distribution. As before, we have that log ( J u ) is cohomolo-gous to a constant function. Since the measure of maximal entropy of the transitiveAnosov diffeomorphism α ( a ) is the equilibrium state of the constant function [17,Theorem 20.1.3], and since the equilibrium state is unique [17, Theorem 20.3.7] weconclude that µ is the measure of maximal entropy for α ( a ). Since λ is the measureof maximal entropy for ρ ( a ) then by conjugation so is the measure φ − ∗ ( λ ) for α ( a ).Hence by uniqueness we have µ = φ − ∗ ( λ ). ⋄ We will show below that the Lyapunov exponents of ( α, µ ) and ( ρ, λ ) are positivelyproportional and that the corresponding coarse Lyapunov foliations are mapped intoeach other by the conjugacy φ .We consider the suspensions ˜ α and ˜ ρ of α and ρ . These are smooth R k actions onthe suspension manifolds S and R of α and ρ . We denote the lifts to the suspensionsof the conjugacy and the invariant measures by ˜ φ , ˜ µ , and ˜ λ . Note that ˜ φ and ˜ φ − are also H¨older continuous with the same exponent γ > φ and φ − .From now on, instead of indexing a coarse Lyapunov by a representative of theclass of positively proportional Lyapunov functionals, we index them numerically.I.e. we write W i instead of W χ , implicitly identifying the finite collection of equiva-lence classes of Lyapunov exponents with a finite set of integers. The next proposi-tion summarizes important properties of the suspension actions. Similar propertieshold for the original Z k actions. Proposition 2.5.
Assume there is an Anosov element in every Weyl chamber. Then (1)
The Lyapunov exponents of ( ˜ α, ˜ µ ) and ( ˜ ρ, ˜ λ ) are positively proportional, andthus the Lyapunov hyperplanes and Weyl chambers are the same. (2) For any coarse Lyapunov foliation W i ˜ α of ˜ α ˜ φ ( W i ˜ α ) = W i ˜ ρ , where W i ˜ α is the corresponding coarse Lyapunov foliation for ˜ ρ . Remark.
We do not claim at this point that the Lyapunov exponents of ( ˜ α, ˜ µ )and ( ˜ ρ, ˜ λ ) (or of different invariant measures for ˜ α ) are equal. Of course if ˜ α is shownto be smoothly conjugate to ˜ ρ then this is true a posteriori. Remark.
In fact, one can show that the same holds for Lyapunov exponents andcoarse Lyapunov foliations of ( α, ν ) for any α -invariant measure ν so, in particular,the Lyapunov exponents of all α -invariant measures are positively proportional andthe coarse Lyapunov splittings are consistent with the continuous one defined inProposition 2.3. Proof : First we observe that the conjugacy ˜ φ maps the stable manifolds of ˜ α tothose of ˜ ρ . More precisely, for any a ∈ R k and any for µ -a.e. x ∈ S we have(3) ˜ φ ( W − ˜ α ( a ) ( x )) = W − ˜ ρ ( a ) ( ˜ φ ( x )) . Indeed, it suffices to establish this for local manifolds, which are characterized by theexponential contraction as in (2). Since ˜ φ is bi-H¨older, it preserves the property thatdist( x n , y n ) decays exponentially, which implies (3). In particular, for any Anosov a ∈ R k and any x ∈ S we have ˜ φ ( W s ˜ α ( a ) ( x )) = W s ˜ ρ ( a ) ( ˜ φ ( x )). Hence the formula for W i ˜ α given in Proposition 2.3 implies (2) once we establish (1).To establish (1) it suffices to show that the oriented Lyapunov hyperplanes of( ˜ α, ˜ µ ) and ( ˜ ρ, ˜ λ ) are the same. Suppose that an oriented Lyapunov hyperplane L of one action, say ˜ α , is not an oriented Lyapunov hyperplane of the other action ˜ ρ .We take an element a close to L in the corresponding positive Lyapunov half-space L + and denote the reflection of a across L by b . We can choose them so that a and b are not separated by any other Lyapunov hyperplane of either action. Then, E − ˜ α ( b ) = E − ˜ α ( a ) ⊕ E , where E is the coarse Lyapunov distribution of ˜ α correspondingto L . Similarly, since we assumed that L + is not a positive Lyapunov half-space for˜ ρ , we have E − ˜ ρ ( b ) ⊆ E − ˜ ρ ( a ) . We conclude that W − ˜ α ( a ) ( W − ˜ α ( b ) but W − ˜ ρ ( a ) ⊇ W − ˜ ρ ( b ) , which contradicts (3) since ˜ φ is a homeomorphism. ⋄ Outline of the proof of Theorem 1.1
Proposition 2.3 shows that coarse Lyapunov foliations for α and ˜ α are well-definedcontinuous foliations with smooth leaves. By Proposition 2.5 they are mapped bythe conjugacy to the corresponding homogeneous foliations for ρ and ˜ ρ . The maingoal is to study the regularity of the conjugacy φ along these foliations.For the most of the proof we consider a coarse Lyapunov foliation W of the suspen-sion action ˜ α . The first major step is to establish smoothness of certain holonomiesbetween leaves of W . The TNS assumption gives the existence of invariant foliations W and W such that T W ⊕ T W ⊕ T W ⊕ T O = T M . Moreover, each T W i ⊕ T W is the stable distribution of some element and, in particular, is integrable. In Section5 we show that the holonomies along W (and along W ) between leaves of W are OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 11 C ∞ . This follows from the existence of an element for which W a fast stable foli-ation inside T W ⊕ T W . To obtain such an element we establish in Section 4 thatthe expansion or contraction of W by an element in the corresponding Lyapunovhyperplane is uniformly slow.The second major step is to establish smoothness of the conjugacy φ along theleaves of the coarse Lyapunov foliation W . For this we introduce in Section 6 themeasurable normal forms for the action on W defined almost everywhere with re-spect to the measure µ = φ − ∗ ( λ ). In Section 7 we show that the smooth holonomiesalong W preserve the normal forms on W . For this we use the semisimplicity as-sumption to split the homogeneous foliation ˜ φ ( W ) into subfoliations correspondingto eigenspaces of ρ . Then we see that holonomies along a particular subfoliationpreserve the normal forms since they commute with an element in R k which fixesthe corresponding eigenspace and contracts W . Since W is the full stable foliationof some element, it is ergodic with respect to µ , and hence the holonomies alonga typical leaf are sufficiently transitive. Using this we show in Section 8 that fora typical leaf W of W and for almost every translation T of the homogeneous leaf φ ( W ), the conjugate map φ − ◦ T ◦ φ : W → W can be obtained as a certain limitof such holonomies. Then this map also preserves the normal forms and therefore issmooth. This yields that φ is C ∞ along W .Since the holonomies between different leaves of W along W and W are smoothand intertwine the restriction of φ to these leaves we obtain that φ is C ∞ along allleaves of W and that the derivatives are continuous transversally. Then standardelliptic theory implies that φ is C ∞ on M .4. Uniform estimates for elements near Lyapunov hyperplane
We consider the suspension actions ˜ α and ˜ ρ of R k on S and R . We fix a Lyapunovhyperplane L ⊂ R k and the corresponding positive Lyapunov half-space L + . Wedenote the corresponding coarse Lyapunov distributions for ˜ α and ˜ ρ by E and ¯ E respectively. Recall that γ > φ and ˜ φ − . Lemma 4.1.
Consider an element b ∈ R k . Let ¯ χ ( b ) be the largest Lyapunov ex-ponent of ˜ ρ ( b ) corresponding to ¯ E and denote χ M = max { , ¯ χ ( b ) /γ } . Let ν be anyergodic invariant measure for ˜ α ( b ) and let χ ν ( b ) be the largest Lyapunov exponentof ( ˜ α ( b ) , ν ) corresponding to the distribution E . Then χ ν ( b ) ≤ χ M Proof : Suppose that χ ν ( b ) > χ M . Let E uu be the distribution spanned by theLyapunov subspaces of ( ˜ α ( b ) , ν ) corresponding to Lyapunov exponents greater than χ M + ε . Then, for some ε > E uu has nonzero intersection with the distribution E .The strong unstable distribution E uu ( x ) is tangent for ν -a.e. x to the correspondingstrong unstable manifold W uu ( x ). Hence the intersection F ( x ) of W uu ( x ) with theleaf W ( x ) of the coarse Lyapunov foliation corresponding to E is a submanifold ofpositive dimension. Take y ∈ F ( x ) and denote y n = ˜ α ( − nb )( y ) and x n = ˜ α ( − nb )( x ). Then x n and y n converge exponentially with the rate at least χ M + ε . Since theconjugacy ˜ φ is γ bi-H¨older it is easy to see thatdist( ˜ φ ( x n ) , ˜ φ ( y n )) = dist( ˜ ρ ( − nb )( x ) , ˜ ρ ( − nb )( y ))decreases at a rate faster than γ χ M . But this is impossible since ˜ φ maps W ( x ) tothe corresponding foliation of the linearization which is contracted by ˜ ρ ( − b ) at arate at most γ χ M . ⋄ Proposition 4.2.
Let L ⊂ R k be a Lyapunov hyperplane and E be the correspondingcoarse Lyapunov distribution for ˜ α . For any ε > there exist η > so that for anyelement b ∈ R k with dist ( b, L ) ≤ η ε there exists C = C ( b, ε ) such that (4) ( Ce εn ) − k v k ≤ k D ( ˜ α ( nb )) v k ≤ Ce εn k v k for all v ∈ E, n ∈ N . Proof : In the proof we will abbreviate ˜ α ( b ) to b . Consider functions a n ( x ) =log k Db n | E ( x ) k , n ∈ N . Since the distribution E is continuous, so are the func-tions a n . The sequence a n is subadditive, i.e. a n + k ( x ) ≤ a n ( b k ( x )) + a k ( x ). TheSubadditive and Multiplicative Ergodic Theorems imply that for every b -invariantergodic measure ν the limit lim n →∞ a n ( x ) /n exists for ν -a.e. x and equals the largestLyapunov exponent of ( b, ν ) on the distribution E .The largest exponent ¯ χ ( b ) of ˜ ρ ( b ) from Lemma 4.1 can be estimated from aboveby c · dist ( b, L ) for some c >
0. Hence we can find η > χ M from Lemma 4.1 is less than ε/ b ∈ R k with dist ( b, L ) ≤ η ε . Then Lemma4.1 implies that lim n →∞ a n ( x ) /n ≤ ε/ x with respect to any b -invariant ergodic measure ν . Thus the exponential growth rate of k Db n | E ( x ) k isless than ε/ b -invariant ergodic measures. Since k Db n | E ( x ) k is continuous,this implies the uniform exponential growth estimate, as in the second inequality in(4) (see [28, Theorem 1] or [27, Proposition 3.4]). The first inequality in (4) can beobtained from the second one for − b . ⋄ Smooth holonomies.
We consider the suspension actions ˜ α and ˜ ρ of R k on S . We fix a Lyapunovhyperplane L ⊂ R k and denote by E and W the corresponding coarse Lyapunovdistribution and foliation for ˜ α on S . These are unique by the TNS hypothesisas there are no negatively proportional Lyapunov exponents. In this section weestablish smoothness of certain holonomies between leaves of W .We first need a technical result on existence of suitable complementary foliations. Lemma 5.1.
For a TNS action α , suppose that every Weyl chamber contains anAnosov element. Then there are ˜ α -invariant distributions E and E such that E ⊕ E ⊕ E ⊕ T O = T S . Moreover, both E i and E i ⊕ E , i = 1 , are the stable OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 13 distribution of some Anosov elements, and hence are tangent to invariant foliationswhich we denote respectively by W i and W i ⊕ W , i = 1 , .Proof : Consider a generic plane P in R k which intersects different Lyapunovhyperplanes in distinct lines L i . Recall that the TNS assumption implies that eachLyapunov hyperplane bounds a unique negative Lyapunov half-space. Thus the L i are naturally oriented, and we can order them cyclically L = L , L , ..., L n . Let m be the index such that − L is between L m and L m +1 . There are two Weyl chambersin the negative Lyapunov half-space L − whose intersections with the plane P border L . By assumption, there exist Anosov elements in these Weyl chambers, which wedenote a and a . Similarly, there are two Weyl chambers across L in the positiveLyapunov half-space L +1 . We denote Anosov elements in these Weyl chambers by c and c . More precisely, if we order the Weyl chambers intersecting P cyclicallyfrom L : C i , i = 1 , ..., n then we can take a ∈ C , a ∈ C m , c ∈ C m +1 , c ∈ C n .Denote the coarse Lyapunov distribution corresponding to L i by E i . Note that E = E . Then one can see that T S = E ⊕ E ⊕ E ⊕ T O , where E := E sc = E ⊕ .... ⊕ E m E sa = E ⊕ .... ⊕ E m = E ⊕ E and E := E sc = E m +1 ⊕ .... ⊕ E n E sa = E m +1 ⊕ .... ⊕ E n ⊕ E = E ⊕ E. Since stable distributions of Anosov elements integrate to invariant foliations, theclaim follows. ⋄ We will show that the holonomies along W i , i = 1 , W are C ∞ .This follows from the existence of an element which contracts W (resp. W ) fasterthan it does W . Proposition 5.2.
In the above notations, for i = 1 , , there exist elements b i ∈ R k such that b i contracts W i faster than it does W , i.e. (5) k D ( ˜ α ( b i )) | E i k < k D ( ˜ α ( − b i )) | E k − ≤ k D ( ˜ α ( b i )) | E k < . Since the faster part of an (un)stable foliation is C ∞ inside of an (un)stable leaf,see for example [14, Proposition 5.1] or [15, Proposition 3.9], we obtain the followingcorollary: Corollary 5.3.
In the above notations, for i = 1 , , the leaves of W i vary smoothlyalong the leaves of W , and the holonomies along W i between leaves of W are C ∞ .Proof : ( of Proposition 5.2.) We use the notations from the proof of Lemma5.1. We will first find an element b ′ close to L which does not expand or contract E much, with uniform control. Then a suitable combination of a i with b ′ will suffice.We consider the case i = 1 and denote a = a , c = c , and F = E ⊕ E . The othercases are similar, and will not be discussed. We have that a uniformly contracts F and c uniformly contracts E , i.e. there exist C , χ > t > k D ( ˜ α ( ta )) v k ≤ C e − χt k v k ∀ v ∈ F, k D ( ˜ α ( tc )) v k ≤ C e − χt k v k ∀ v ∈ E Since E is a continuous distribution on S and S is compact, a contracts E by atmost sup x ∈S k da − | E ( x ) k . Hence there is a fastest contraction rate χ ′ for a on E such that for some c > t > k D ( ˜ α ( ta )) v k ≥ c e − χ ′ t k v k ∀ v ∈ E Since F ⊃ E , equations (6) and (7) imply that χ ′ > χ .Let b ′ = ra + (1 − r ) c , 0 < r <
1, be a convex combination of a and c . Note thatby (6) any such b ′ uniformly contracts E :(8) k D ( ˜ α ( tb ′ )) v k ≤ C e − χt k v k ∀ v ∈ E , ∀ t > . We will find an element satisfying (5) in the form b = t ( b ′ + sa ), where t > s > ε > b ′ so that it is in L − and sufficiently close to L so that Proposition 4.2 applies and so b ′ contracts E veryslowly. Then equations (6), (7), (8) yield that there exists K > t > k D ( ˜ α ( b )) v k ≤ Ke − ( χ + sχ ) t k v k ∀ v ∈ E , and K − e − ( sχ ′ + ε ) t k v k ≤ k D ( ˜ α ( b )) v k ≤ Ke − ( sχ − ε ) t k v k ∀ v ∈ E. We conclude that b will satisfy (5) for sufficiently large t if we choose ε and s so that sχ ′ + ε < χ + sχ while sχ − ε >
0. This is equivalent to εχ < s < χ − εχ ′ − χ and hence we can choose such s if ε is sufficiently small. ⋄ Normal forms
We consider the suspension action ˜ α of R k on S . We fix a Lyapunov hyperplane L ⊂ R k and denote by E and W the corresponding coarse Lyapunov distributionand foliation for ˜ α .In this section we study properties of the action along the leaves of W and in-troduce smooth coordinate changes along the leaves of W with respect to whichthe elements act as certain polynomials. This method was introduced to the studyof local rigidity of higher rank abelian actions in [20] and uses the nonstationarynormal forms of smooth contractions developed in [8, 7]. In contrast to the caseof small perturbations of algebraic actions considered in [20], the action ˜ α may nothave the so-called “narrow band” property. Instead of uniform growth estimatesgiven by the narrow Mather spectrum, we have to use nonuniform estimates givenby the Multiplicative Ergodic Theorem for the measure µ . Therefore, the coordinatechanges will vary on S not continuously but measurably.Let a be an element in the negative Lyapunov half-space L − ⊂ R k , so that f = ˜ α ( a ) contracts W . We will view it as a measure-preserving system ( f, µ ).Its action along W , f : W ( x ) → W ( f x ), defines an extension Φ : S × R m → S × R m OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 15 of f , where m = dim W . Indeed, the leaf W ( x ) can be smoothly identified with thetangent space E ( x ), and the distribution E can always be measurably trivializedon a set of full measure. The extension Φ a preserves the zero section and acts by C ∞ diffeomorphisms in the fibers. In other words, Φ a can be written in coordinates( x, t ) ∈ S × R m as Φ a ( x, t ) = ( f ( x ) , F x ( t ))where F x (0) = 0 and F is C ∞ in t . We will allow coordinate changes which aremeasurable in x , preserve each fiber R mx , fix the origin, are C ∞ in each fiber, andhave tempered logarithms of all derivatives of all orders at the zero section. Wewill call such coordinate changes admissible . Recall that a real-valued function ϕ iscalled tempered with respect to the action ˜ α if lim b →∞ k b k − ϕ ( ˜ α ( b ) x ) = 0 for µ -a.e. x .The derivatives in the t variable at the zero section define a linear extension of f ,which we will denote by D F x and call the derivative extension. Note that D F x arebounded functions on S and that this extension has negative Lyapunov exponents.Let χ , . . . χ l be the different Lyapunov exponents of the derivative extension and m , . . . , m l be their multiplicities. Represent R m as the direct sum of the spaces R m i , . . . , R m l and let ( t , . . . , t l ) be the corresponding coordinate representation ofa vector t ∈ R m . Let P : R m → R m ; ( t , . . . , t l ) ( P ( t , . . . , t l ) , . . . , P l ( t , . . . , t l ))be a polynomial map preserving the origin. We will say that the map P is of subres-onance type if it contains only such homogeneous terms in P i ( t , . . . , t l ) with degreeof homogeneity s j in the coordinates of t j , i = 1 , . . . , l for which the subresonancerelation χ i ≤ P j = i s j χ j holds. There are only finitely many subresonance relationsand it is known [7, 8] that polynomial maps of the subresonance type with invertiblederivative at the origin generate a finite-dimensional Lie group. We will denote thisgroup by SR χ . Proposition 6.1.
There exists an admissible coordinate change in
S × R m whichtransforms the extensions Φ a for all a ∈ L − to extensions Ψ a of the subresonancenormal form Ψ( x, t ) = ( f ( x ) , P x ( t )) where for almost every x ∈ X, P x ∈ SR χ .Moreover, this admissible coordinate change transforms into such normal formany extension Γ( x, t ) = ( g ( x ) , G x ( t )) by C ∞ diffeomorphisms preserving the zerosection of a non-singular transformation g of ( S , µ ) which commutes with Φ a forsome a ∈ L − .Proof : We note that since E is a coarse Lyapunov distribution, all Lyapunovexponents of ˜ α corresponding to E are, by definition, positively proportional. There-fore, the extensions Φ a for all a ∈ L − are contractions with the same subresonancerelations. The existence of an admissible coordinate change for a single a ∗ ∈ L − is given by Theorem 6.1 in [12]. Since Φ a commutes with Φ a ∗ , the “centralizer theo-rem” [12, Theorem 6.3] yields that this coordinate change brings any other Φ a , for a ∈ L − , to the subresonance normal form of Φ a ∗ . The coincidence of resonancesimplies that this normal form is also the normal form for Φ a . Then the “centralizertheorem” can be applied to this coordinate change with any a ∈ L − and yields thesecond part of the proposition. ⋄ Commuting holonomies
Let W be a coarse Lyapunov foliation as in Section 5 with a complementary foli-ation W as in Lemma 5.1. To simplify notations in Sections 7 and 8 we will denotethe corresponding foliations for the algebraic action ˜ ρ by W ∗ and W ∗ respectively.In this section we will study the holonomies along W between leaves of W . Whilein a general setting holonomy along a foliation is only a locally defined operation,in our setting holonomies are realized by global homeomorphisms. Before specifyingthis we introduce some notations and describe the algebraic foliations W ∗ and W ∗ .We have two actions ˜ α and its linearization ˜ ρ on the suspension manifold S .Recall that S is a homogeneous space S/ Λ where S = R k ⋉ N is a solvable Liegroup and Λ = Z k ⋉ Γ is a lattice in S . A left coset foliation is the foliation definedby orbits of some subgroup D < S . The foliations W ∗ and W ∗ , as well as othercoarse Lyapunov and stable foliations for the ˜ ρ on S , are left coset foliations. Thisis most easily seen at the level of Lie algebras. Let s be the Lie algebra of S and n the Lie algebra of N . We can identify the tangent bundle of S with S × s . Thefibration N/ Γ → S → R k / Z k of N/ Γ defines a foliation whose tangent bundle isgiven by n in this identification. Since ˜ ρ is the suspension of the action ρ by affineautomorphisms of N/ Γ any “dynamical” foliation as above is tangent to an invariantdistribution given by a subspace d ⊂ n , which by integrability is a Lie subalgebra of n . This makes the corresponding foliation into a left coset foliation for the subgroup D < N such that
Lie ( D ) = d . For the coarse Lyapunov foliation W ∗ , and for thecomplementary foliation W ∗ we will denote the corresponding nilpotent groups by W and W .Recall that by Lemma 5.1 W ∗ and W ∗ subfoliate the leaves of W ∗ ⊕ W ∗ , whichis a stable foliation for ˜ ρ . Moreover, on each leaf of this foliation they form a globalproduct structure. This can be seen on the universal cover, which for the algebraicaction on N/ Γ can be identified with the Lie algebra n . We choose any element b ∈ W and denote the translation action of b on S by L b ( x ) = b · x = bx . Thenfor any such b and any x in S the holonomy along W ∗ is a diffeomorphism between W ∗ ( x ) and W ∗ ( bx ), which we denote by h ∗ b,x .Similarly, for any b in W and any x in S we denote by h b,x the holonomy along W between W ( x ) and W ( bx ). Since the conjugacy ˜ φ maps W to W ∗ and W to OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 17 W ∗ we see that h b,x is a global homeomorphism and h ∗ b,x ◦ ˜ φ = ˜ φ ◦ h b,x . Moreover, h b,x is a diffeomorphism by Corollary 5.3.In order to use Proposition 6.1 we can use the holonomies h ∗ b,x and h b,x to definebundle maps in the following manner. We take a bundle S × W with base S andfiber W s , which we think of as the leaf of W ∗ or W through s . For the leaves of W ∗ we have a natural identification of W s with W ∗ ( s ) given by left translations: w w · s . For W we fix some smooth identification that depends continuously on s in C ∞ topology on compact subsets of W . The holonomy h ∗ b can now be viewedas a bundle map h ∗ b : S × W → S × W covering the left translation L b in the base,where h ∗ b ( x, w ) = h ∗ b,x ( w ).Similarly, we define h b via the equation h b ( x, w ) = h b,x ( w ) and h b is a bundle map h b : S× W → S× W which is smooth along the fibers and covers the homeomorphism φ − ◦ L b ◦ φ of S . Note that this homeomorphism preserves the invariant measure µ on S since L b preserves the Lebesgue measure λ = ˜ φ ∗ ( µ ). Since the nilpotent group W is diffeomorphic to R m , we are in the setup of Proposition 6.1. The actions ˜ α and ˜ ρ also lift naturally to the corresponding actions on the bundle S × W . Slightlyabusing notations we will denote the lifts by the same letters. Since we do not knowthe smoothness of W , we can only say that the lift of ˜ α is smooth along the fibers.Note that the natural extension of ˜ φ to S × W conjugates the lifts of the actions aswell as the holonomy maps h b and h ∗ b .We will use the algebraic structure of ˜ ρ to show that h ∗ b commute with certainelements of ˜ ρ ( R k ) and then use this to conclude that h b commute with certainelements of ˜ α ( R k ). The main goal of this section is to prove the following theorem. Theorem 7.1.
If the action ρ is semisimple, then for every b ∈ W the maps h b forall b ∈ W preserve, µ -almost everywhere, a fixed normal form along leaves of W .Proof : We begin by finding subfoliations of W for which the holonomy commuteswith some ˜ α ( v ), v ∈ R k , contracting W . To do this we will work with the algebraicaction ˜ ρ . Recall that T W ∗ splits as a sum of coarse Lyapunov distributions ⊕ E j . Let E ′ be one of these distributions and L ′ be the corresponding Lyapunov hyperplane.Let v be any element of L ′ for which ˜ ρ ( v ) contracts W ∗ . Then ˜ ρ ( v ) acts isometricallyon a certain foliation H ∗ v which is defined as the orbits of the action of some subgroup H v in W . Since ˜ ρ ( v ) is semisimple by the assumption, H ∗ v is in fact the full coarseLyapunov foliation of ˜ ρ corresponding to L ′ . (If the derivative of ˜ ρ ( v ) on E ′ hadJordan blocks, this would be a strict subfoliation.) Now we can decompose the Liealgebra of H v into the irreducible subspaces of the rotation defined by taking theskew symmetric part of ˜ ρ ( v ). We denote the resulting Lie subgroups of H v by H v,i . Lemma 7.2.
For any element v ∈ L ′ there are real numbers t i > such that forany b ∈ H v,i the map h ∗ b commutes with ˜ ρ ( t i v ) . Proof : A suitable multiple t i v of v commutes with the group H v,i . Hencetranslations by elements of H v,i commute with ˜ ρ ( t i v ). Then the holonomy h ∗ b willalso commute with ˜ ρ ( t i v ) since it agrees with holonomy along the foliation H ∗ v,i . ⋄ Since ˜ φ conjugates the actions and the holonomies this lemma yields that forany b ∈ H v,i the map h b commutes with ˜ α ( t i v ). By making different choices of theLyapunov hyperplane L ′ , v ∈ L ′ , and i , we can arrange so that the groups H v,i generate W . Proposition 6.1 implies that there is a common normal form for h b for all b in all H v,i . Therefore, h b with any b in W , which is a composition of mapsof this form, also preserves this normal form. This completes the proof of Theorem7.1. ⋄ Now we give a more detailed description of the algebraic holonomies h ∗ b . As acorollary we describe certain limits of the maps h ∗ b and h b which will be used in thenext section. Proposition 7.3.
For any b ∈ W and any x ∈ S , the holonomy h ∗ b,x : W ∗ ( x ) → W ∗ ( bx ) is equivariant with respect to the action of W along leaves of W ∗ .Proof : We need to show that the holonomy along W ∗ commutes with the actionof W along leaves of W ∗ . First we observe that W normalizes W . To see this wenote that there is a subgroup W ′ = WW in N . This is the group that correspondsto the foliation W ∗ ⊕W ∗ as in Lemma 5.1. We denote the Lie algebras of W and W by w and w . To conclude that W is normal inside W ′ we choose an element s ∈ R k for which ˜ ρ ( s ) acts isometrically on w and contracts exactly w . This ispossible by the construction of W ∗ and W ∗ in Lemma 5.1. Then any bracket [ v, u ]where u ∈ w and v ∈ w is contracted by ˜ ρ ( s ) and hence must be in w .Now let a ∈ W and b ∈ W . For any x ∈ S we can write abx = aba − ax . By thenormalization we have aba − ∈ W . Hence the point abx is both in W ∗ ( bx ) and in W ∗ ( ax ). This shows that h ∗ b,x ( ax ) = abx for any x ∈ S and b ∈ W and proves thatthe holonomy h ∗ b,x from W ∗ ( x ) to W ∗ ( bx ) commutes with the action of W . ⋄ Corollary 7.4.
Suppose that for some elements b n ∈ W and some point x ∈ S the sequence b n x converges to a point y in W ∗ ( x ) . Then the holonomy maps h ∗ b n ,x converge to the diffeomorphism T ∗ x,y : W ∗ ( x ) → W ∗ ( x ) given by T ∗ x,y ( ax ) = ay forany a ∈ W . Consequently, the holonomy maps h b n ,x converge to the homeomorphism ˜ φ − ◦ T ∗ x,y ◦ ˜ φ of the leaf W ( ˜ φ − ( x )) uniformly on compact sets.Proof : By Proposition 7.3, if a ∈ W then h ∗ b n ,x ( ax ) = ah ∗ b n ,x ( x ) = a ( b n x ),which converges to ay as desired. The first claim now follows since W ∗ ( x ) = W x .Conjugating by ˜ φ gives the second claim. ⋄ OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 19
Remark:
It is not clear that the limit of the h ∗ b n ,x can be realized as a holonomy ofany kind along any leaf from W ∗ ( x ) to W ∗ ( y ).8. Limiting Argument
The main goal of this section is to prove the following proposition, which we willthen use to complete the proof of Theorem 1.1. We retain the notations of theprevious section.
Proposition 8.1.
For µ -almost every x ∈ S there are smooth transitive actions of W on the leaves W ( x ) and W ∗ ( x ) which are intertwined by the conjugacy ˜ φ .Proof : For any given x ∈ S we can naturally identify W with W ∗ ( x ) = W x by w wx . In this identification, we take the desired transitive action of W on W ∗ ( x ) to be the action by right translations. Corollary 7.4 means that the limitsof the holonomy maps h ∗ b n ,x are exactly of this form. In fact, since W ∗ ( x ) is densein the corresponding N/ Γ-fiber of the suspension as a full stable foliation of Anosovelement, one can see that any right translation can be obtained as such a limit.While we will not use it directly, this provides motivation for the argument. Thedesired action of W on W ∗ ( x ) is obtained by conjugating by ˜ φ . This action is apriori only by homeomorphisms and the goal is to prove that it is smooth. For thiswe will study the limits of the holonomy maps h b n ,x .We consider Lusin sets where the measurable normal form on the leaf W ( x )depends continuously on x . Let Λ ′ m be an increasing sequence of such sets with µ (Λ ′ m ) →
1. Let Λ m be the set of density points of Λ ′ m , then µ (Λ m ) →
1. Thenthere exists a subset X ⊂ Λ = ∪ Λ m with µ ( X ) = 1 such that for all x ∈ X theintersection W ( x ) ∩ Λ has full measure with respect to the conditional measure of µ on W ( x ).Fix any x ∈ X . Then for almost every y in W ( x ) with respect to the conditionalmeasure x and y belong to some Λ m . We pick a sequence b n of elements in W withthe following properties:(1) x n = b n x → y (2) x n ∈ Λ m .To find such b n we use the fact that y is a density point of Λ m and the fact that W acts ergodically with respect to µ on the corresponding N/ Γ-fiber of the suspension.The ergodicity follows since the foliation W of N/ Γ is a full stable foliation ofsome Anosov element of α and hence is uniquely ergodic by Bowen and Marcus [3].Alternately, since the push forward of µ by φ is Lebesgue, the ergodicity can bechecked on the algebraic side using the work of Auslander, Hahn, and Green [1].Each map h b n ,x is smooth and preserves the normal forms at x and x n . ByCorollary 7.4 the sequence h b n ,x converges to a homeomorphism T x,y : W → W conjugate by ˜ φ to the translation T ∗ ˜ φ ( x ) , ˜ φ ( y ) of W ∗ ( ˜ φ ( x )). Since the normal formcoordinates depend continuously on the base point in Λ m and the maps h b n ,x in these coordinates belong to a fixed Lie group, the limit T x,y is smooth. Recall that thepush forward of µ by ˜ φ is the Lebesgue measure λ and hence the conditional measureof µ on W ( x ) is mapped by ˜ φ to the conditional measure of λ on W ∗ ( ˜ φ ( x )), whichis equivalent to volume on W ∗ ( ˜ φ ( x )) = W ˜ φ ( x ). We conclude that for almost everyelement of W the corresponding translation is conjugate by ˜ φ to a diffeomorphismof W ( x ). Hence the subgroup of W that acts by diffeomorphisms of W ( x ) has fullmeasure and must be the whole W by the next lemma. It now follows from [24,Section 5.1, Corollary] that the action of W on W ( x ) is smooth. This completesthe proof of Proposition 8.1. ⋄ Lemma 8.2.
Let G be a Lie group. Then any subgroup H of full measure is G .Proof : If not then the distinct cosets of H in G are disjoint sets of full measurewhich is impossible. ⋄ Remark:
It is possible to prove that G is smooth along a generic leaf of W us-ing older methods involving returns along Weyl chamber walls in R k instead ofholonomies. However, one cannot obtain uniformity in estimates this way nor com-plete the proof below without using holonomies. End of Proof of Theorem 1.1 : We need to show that φ is a diffeomorphism. Itwill be easier to work with φ − as we will employ certain elliptic operators definedby right invariant vector fields to prove smoothness of φ − .Proposition 8.1 implies that for any coarse Lyapunov foliation ˜ φ − intertwinestransitive C ∞ group actions on typical leaves W ( x ) and W ∗ ( x ) in the suspension S .This yields that, for a typical x in M = N/ Γ, φ − intertwines transitive C ∞ groupactions on W ( x ) and W ∗ ( x ). Hence φ − is C ∞ along W ∗ ( x ).We claim that φ − is C ∞ along all leaves of W ∗ and that all its derivatives along theleaves are continuous on M . This follows from the fact that T M = T W ⊕ T W ⊕ T W and that the holonomies between different leaves of W along W and W are smoothand intertwine the restriction of φ − to these leaves.We can now finish the proof quickly. We know that φ − is smooth along the coarseLyapunov foliations with continuous dependence of the derivatives. This simplysays that derivatives of all orders exist and are continuous for each right invariantvectorfield tangent to a coarse Lyapunov foliations (while mixed derivatives mayfail to exist). Pick a basis of such vectofields X i . Then X l := P i X li for any l isan elliptic operator of order 2l. It follows that X l ( φ − ) is smooth for all l . Henceby elliptic theory, φ − is C ∞ . We refer to [5, Section 7.1] e.g. for a more detaileddiscussion of this elliptic theory argument. OTALLY NON-SYMPLECTIC ANOSOV ACTIONS 21
It remains to show that φ − is a diffeomorphism. Since φ − is already a homeo-morphism, this follows once we show that the differential of φ − is everywhere non-degenerate. This follows easily from Proposition 2.4. Indeed, we have µ = φ − ∗ ( λ )and µ has smooth positive density. ⋄ Totally reducible actions and examples.
Here we will prove Corollary 1.2. By the proposition below, this is immediatefrom Theorem 1.1.Recall that an algebraic Z k action on a torus is called irreducible if there is norational invariant subtorus, and totally reducible if every rational invariant subtorushas a rational invariant complement.Given a nilmanifold N/ Γ, there is a maximal toral quotient T d obtained by taking N/ [ N, N ]Γ. Any action by automorphisms on N/ Γ descends to an action on T d ,which we refer to as the maximal toral quotient action . We say that an algebraic Z k action on N/ Γ is totally reducible if the maximal toral quotient action is totallyreducible and there is a Z k invariant complement to [ n , n ] in the Lie algebra n of N .We call an action by affine automorphisms of a nilmanifold totally reducible if thefinite index subgroup that acts by automorphisms is totally reducible.It is easy to see that semisimple actions are totally reducible. Proposition 9.1.
A totally reducible Z k action on a nilmanifold is semisimple.Proof : First we consider an irreducible torus action. Let A be a toral auto-morphism, i.e. an integral matrix. The characteristic polynomial of A splits over Q as Q P i ( X ) d i . Then the kernel E ( A ) of Q P i ( A ) is the subspace spanned by theeigenspaces of A . It is rational as the kernel of a rational operator.If a collection A i of toral automorphisms commute then E ( A ) is invariant under A . Consider the restriction B of A to E ( A ) Then E ( B ) is nontrivial, andcontained in E ( A ) ∩ E ( A ). Inductively we see that ∩ E ( A i ) is nontrivial. Thus weget a nontrivial rational subspace invariant under all A i . This defines an invariantproper subtorus unless all A i are semisimple. Hence irreducible torus actions aresemisimple.Considering irreducible components of totally reducible torus actions it followseasily that they are also semisimple.Finally consider a totally reducible action on a nilmanifold. Then the maximaltoral quotient action is totally reducible and hence semisimple. This implies thatthe action on the invariant complement R d to [ n , n ] is semisimple. Since joint eigen-vectors for Z k span R d , their brackets, which are also eigenvectors span n . Thereforethe action is semisimple. ⋄ We briefly describe many examples of totally irreducible Anosov actions on nil-manifolds. These examples are more general variants of examples constructed byQian in [26]. Let T d be a torus with an Anosov algebraic semisimple Z k action. Theaction lifts to the vector space R d . Let N = N k ( R d ) be the k -step free nilpotentLie group generated by R d . (It is somewhat more typical to define this at the levelof Lie algebras, but the meaning is clear as long as we assume N k ( R d ) is simplyconnected.) The Z k action on R d extends canonically to a Z k action on N k ( R d ) andpreserves the obvious rational structure on that group. This implies that we have awell-defined Z k action on N/ Γ where Γ is a lattice in N .It is easy to check that generically this construction takes an Anosov Z k action on T d and lifts it to an Anosov action on N/ Γ. An Anosov automorphism A of T d liftsto an Anosov automorphism of N/ Γ as long as no product of length at most k ofeigenvalues of A has modulus one. It is straightforward to construct many exampleswhich are also T N S using similar algebraic condition on eigenvalues.We remark that the hypothesis of Theorem 1.1 are necessary for our argument asthere are examples for which the commuting holonomies are not ergodic.
Example 9.1.
Take a semisimple Anosov linear action of Z k on T d , we can define anaction on T d by letting A ∈ Z k act by A ( x, y ) = ( Ax, Ay + x ). It is straightforwardto check that for examples of this kind, the commuting holonomies are not ergodic. References [1] L. Auslander, L. Green, L., F. Hahn.
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