How typical are pathological foliations in partially hyperbolic dynamics: an example
aa r X i v : . [ m a t h . D S ] J un HOW TYPICAL ARE PATHOLOGICAL FOLIATIONS INPARTIALLY HYPERBOLIC DYNAMICS: AN EXAMPLE
ANDREY GOGOLEV
Abstract.
We show that for a large space of volume preserving partiallyhyperbolic diffeomorphisms of the 3-torus with non-compact central leaves thecentral foliation generically is non-absolutely continuous. Introduction
Let M be a smooth Riemannian manifold. In this paper we will consider con-tinuous foliations of M with smooth leaves. A continuous foliation W with smoothleaves W ( x ) , x ∈ M , is a foliation given by continuous charts whose leaves aresmoothly immersed and whose tangent distribution T W is continuous on M . Rie-mannian metric induces volume m on M as well as volume on the leaves of W .Following Shub and Wilkinson [SW00] we call such foliation W pathological if thereis a full volume set on M that meets every leaf of the foliation on a set of leaf-volumezero. According to Fubini Theorem, smooth foliations cannot be pathological, butcontinuous foliations might happen to be pathological. This phenomenon naturallyappears for central foliations of partially hyperbolic diffeomorphisms and is alsoknown as “Fubini’s nightmare.” A diffeomorphism f is called partially hyperbolicif the tangent bundle T M splits into a Df -invariant direct sum of an exponentiallycontracting stable bundle, an exponentially expanding unstable bundle and a cen-tral bundle of intermediate growth (precise definitions appear in the next section).The first example of a pathological foliation was constructed by Katok and ithas been circulating in dynamics community since the eighties. Katok suggestedto consider one parameter family { A t , t ∈ R / Z } of area-preserving Anosov diffeo-morphisms C -close to a hyperbolic automorphism A of the 2-torus. By Hirsch-Pugh-Shub Theorem diffeomorphism F ( x, t ) = ( A t ( x ) , t ) is partially hyperbolicwith uniquely integrable central distribution. Then, under certain generic condi-tions (the metric entropy or periodic eigendata of A t should vary with t ) on path A t , one can show that the central foliation by embedded circles is pathological.See [Pes04], Section 7.4, or [HassP06], Section 6, for detailed constructions withproofs.A version of above construction on the square appeared in expository paper byMilnor [Mil97]. Milnor remarks that a different version of the construction, basedon tent maps, has also been given by Yorke.Shub and Wilkinson [SW00] came across the same phenomenon when lookingfor volume preserving non-uniformly hyperbolic systems in the neighborhood of F : ( x, t ) ( A ( x ) , t ). They have showed existence of C -open set of diffeomor-phisms in the C -neighborhood of F with non-zero central exponent. Then one canargue that the central foliation is pathological using the following “Ma˜n´e’s argu-ment”. By Oseledets’ Theorem the set of Lyapunov regular points has full volume. If any central leaf intersected the set of regular points by a set of positive Lebesguemeasure, then it would increase exponentially in length under the dynamics. Butthe lengths of central leaves are uniformly bounded.Work [SW00] was further generalized by Ruelle [Ru03]. Ruelle and Wilkin-son [RW01] also showed that conditional measures are in fact atomic. Case ofhigher dimensional central leaves was considered by Hirayama and Pesin [HirP07].They showed that central foliation is not absolutely continuous if it has compactleaves and the sum of the central exponents is nonzero on a set of positive measure.This work is devoted to the study of pathological foliations with one-dimensionalnon-compact leaves. Consider a hyperbolic automorphism L of the 3-torus T witheigenvalues ν , µ and λ such that ν < < µ < λ . One can view L as a partiallyhyperbolic diffeomorphism. It was noted in [GG08] and independently in [SX08]that for a small C -open set in the neighborhood of L “Ma˜n´e’s argument” can beapplied to show that corresponding central foliations are pathological. In this paperwe apply a completely different approach to show that there is an open and denseset U of a large C -neighborhood of L in the space of volume preserving partiallyhyperbolic diffeomorphisms such that all diffeomorphisms from U have pathologicalcentral foliations. This result confirms a conjecture from [HirP07].
Acknowledgement.
The author is grateful to Boris Hasselblatt and AnatoleKatok for listening to the preliminary version of the proof of the result. The authorwould like to thank the referees for useful feedback.2.
Preliminaries
Here we introduce all necessary notions and some standard tools that we needfor precise formulation of the result and the proof. The reader may consult [Pes04]for an introduction on partially hyperbolic dynamics.
Definition . A diffeomorphism f is called Anosov if there exists a Df -invariantsplitting of the tangent bundle T M = E sf ⊕ E uf and constants λ ∈ (0 ,
1) and
C > n > k Df n v k ≤ Cλ n k v k , v ∈ E s and k Df − n v k ≤ Cλ n k v k , v ∈ E u . Definition . A diffeomorphism f is called partially hyperbolic if there exists a Df -invariant splitting of the tangent bundle T M = E sf ⊕ E cf ⊕ E uf and positive constants ν − < ν + < µ − < µ + < λ − < λ + , ν + < < λ − , and C > n > C ν n − k v k ≤ k D ( f n )( x ) v k ≤ Cν n + k v k , v ∈ E sf ( x ) , C µ n − k v k ≤ k D ( f n )( x ) v k ≤ Cµ n + k v k , v ∈ E cf ( x ) , C λ n − k v k ≤ k D ( f n )( x ) v k ≤ Cλ n + k v k , v ∈ E uf ( x ) . The following definition is equivalent to the above one. We will switch betweenthe definitions when convenient.
Definition . A diffeomorphism f is called partially hyperbolic if there exists aRiemannian metric on M , a Df -invariant splitting of the tangent bundle T M = ATHOLOGICAL FOLIATIONS 3 E sf ⊕ E cf ⊕ E uf and positive constants ν − < ν + < µ − < µ + < λ − < λ + , ν + < < λ − ,such that ν − k v k ≤ k Df ( x ) v k ≤ ν + k v k , v ∈ E sf ( x ) ,µ − k v k ≤ k Df ( x ) v k ≤ µ + k v k , v ∈ E cf ( x ) ,λ − k v k ≤ k Df ( x ) v k ≤ λ + k v k , v ∈ E uf ( x ) . The distributions E sf , E cf and E uf are continuous. Moreover, distributions E sf and E uf integrate uniquely to foliations W sf and W uf . When it does not lead to aconfusion we drop dependence on the diffeomorphism. By m W σ ( · ) or m σ we denoteinduced Riemannian volume on the leaves of W σ , σ = s, c, u . Induced volume onother submanifolds such as transversals to a foliation will be denoted analogouslywith appropriate subscript.We write d for the distance induced by the Riemannian metric and d σ ( · , · ) forthe distance induced by the restriction of the Riemannian metric to T W σ . Ifexpanding foliation W u is one-dimensional then it is convenient to work with thepseudo-distance ˜ d u ( · , · ) that is very well adapted to the dynamics. Let D uf ( x ) = k Df ( x ) (cid:12)(cid:12) E uf ( x ) k and ρ x ( y ) = Y n ≥ D uf ( f − n ( x )) D uf ( f − n ( y )) . This infinite product converges and gives a continuous positive density ρ x ( · ) on theleaf W u ( x ). Define pseudo-distance ˜ d u by integrating density ρ x ( · )˜ d u ( x, y ) = Z yx ρ x ( z ) dm W u ( x ) ( z ) . Obviously, pseudo-distance is not even symmetric, still it is useful for computationsas it satisfies the formula ˜ d u ( f ( x ) , f ( y )) = D uf ( x ) ˜ d u ( x, y )verified by the following simple computation˜ d u ( f ( x ) , f ( y )) = Z f ( y ) f ( x ) ρ f ( x ) ( z ) dm W u ( f ( x )) ( z )= Z yx ρ f ( x ) ( f ( z )) D uf ( z ) dm W u ( x ) ( z )= Z yx D uf ( x ) D uf ( z ) ρ x ( z ) D uf ( z ) dm W u ( x ) ( z ) = D uf ( x ) ˜ d u ( x, y ) . A compact domain inside a leaf W σ ( x ) of a foliation W σ will be called plaque and will be denoted by P σ . We shall also write P σ ( x ) when we need to indicatedependence on the point.Given a transversal T to W , consider a compact domain X which is a union ofplaques of W , that is, X = ∪ x ∈ T P ( x ). Then by Rokhlin’s Theorem there exists ATHOLOGICAL FOLIATIONS 4 a unique system of conditional measures µ x , x ∈ T , such that for any continuousfunction ϕ on X Z X ϕdm X = Z T Z P ( x ) ϕdµ x d ˆ m, where ˆ m is projection of m X to T . Definition . Foliation W is called absolutely continuous with respect to the volume m if for any T and X as above the conditional measures µ x have L densities withrespect to the volume m P ( x ) for ˆ m a. e. x .Now consider a compact domain X as above and two transversal T and T sothat X = ∪ x ∈ T P ( x ) = ∪ x ∈ T P ( x ) with the same system of plaques. Then theholonomy map p : T → T along W is a homeomorphism. Definition . Foliation W is called transversally absolutely continuous if any holo-nomy map p as above is absolutely continuous, that is, p ∗ m T is absolutely contin-uous with respect to m T .Transverse absolute continuity is a stronger property than absolute continuity.Stable and unstable foliations of Anosov and partially hyperbolic diffeomorphismsare known to be transversally absolutely continuous.3. Formulation of the result
Let L be a hyperbolic automorphism of 3-torus T with positive real eigenvalues ν , µ and λ , ν < < µ < λ . Observe that L can be viewed as a partially hyperbolicdiffeomorphism with L -invariant splitting T T = E sL ⊕ E wuL ⊕ E suL , where “wu” and“su” stand for “weak unstable” and “strong unstable”.Consider the space Diff rm ( T ) of C r , r ≥
2, diffeomorphisms of T that preservevolume m . Let U ⊂
Diff rm ( T ) be the set of Anosov diffeomorphisms conjugateto L via a conjugacy homotopic to identity and also partially hyperbolic. It isknown that U is C -open ( e.g., see [Pes04], Theorem 3.6). Given f in U denoteby E sf ⊕ E wuf ⊕ E suf corresponding f -invariant splitting. According to [BBI09]distributions E wuf , E sf ⊕ E wuf and E uf = E wuf ⊕ E suf integrate uniquely to invariantfoliations W wu , W s ⊕ wu and W u . It is known that W s and W u are C and W su is C when restricted to the leaves of W u (see, e.g., [Hass94, PSW97]). We shall needthe following statement that shows that the structure of weak unstable foliation isessentially linear. Proposition 1.
Let f ∈ U and let h f be the conjugacy to the linear automorphism— h f ◦ f = L ◦ h f . Then h f ( W wuf ) = W wuL . The proof will be given in the appendix.
Theorem A.
There is a C -open and C r -dense set V ⊂ U such that f ∈ V if andonly if the central foliation W wu is non-absolutely continuous with respect to thevolume m .Remark. Since we know that W u is C the latter is equivalent to W wu being non-absolutely continuous on almost every plaque of W u with respect to the inducedvolume on the plaque. ATHOLOGICAL FOLIATIONS 5
Now we describe set V . Given f ∈ U and given a periodic point x of period p let λ su ( x ) = k Df p ( x ) (cid:12)(cid:12) E suf ( x ) k /p . Then set V can be characterized as follows. V = { f ∈ U : there exist periodic points x and y with λ su ( x ) = λ su ( y ) } . Proposition 2.
U\V = { f ∈ U : for any periodic point x λ su ( x ) = λ } . We defer the proof to the appendix.4.
Related questions
Our result does not give any information about the structure of singular condi-tional measures.
Question 1.
Given f ∈ V , what can one say about singular conditional measures on W wu ? Are they atomic? What can be said about Hausdorff dimension of conditionalmeasures? It seems that our method can be generalized for analysis of central foliationof partially hyperbolic diffeomorphisms in a C neighborhood of F : ( x, t ) ( A ( x ) , t ). Question 2.
Is it true that a generic perturbation of F has non-absolutely con-tinuous central foliation? Can one give explicit conditions in terms of stable andunstable Lyapunov exponents of periodic central leaves for non-absolute continuity? It would be interesting to generalize Theorem A to the higher dimensional set-ting. Namely, let L be an Anosov automorphism that leaves invariant a partiallyhyperbolic splitting E sL ⊕ E wuL ⊕ E suL , where E wuL ⊕ E suL is the splitting of the un-stable bundle into weak and strong unstable subbundles. Let n , n and n be thedimensions of E sL , E wuL and E suL respectively. Let U be a small C neighborhoodof L in the space of volume preserving diffeomorphisms. Question 3.
Is it possible to describe the set { f ∈ U : W wu is not absolutely continuous } in terms of strong unstable spectra at periodic points in higher dimensional setting? It will become clear from the discussion in the next section that the value of n is not important. Also it seems likely that our approach works in the case when n > n = 1, and gives a result analogous to Theorem A (the author doesnot claim to have done this).The picture gets much more complicated when n >
1. It is possible that themajor link in our argument( W wu is Lipschitz inside W u ) ⇔ ( W wu is absolutely continuous inside W u )is no longer valid in this setting. However it is not immediately clear how toconstruct a counerexample. ATHOLOGICAL FOLIATIONS 6 Outline of the proof
Clearly V is C -open. Given a diffeomorphism f ∈ U\V we can compose it witha special diffeomorphism h that is C r -close to identity and equal to identity outsidea small neighborhood of a fixed point so that strong unstable eigenvalue of f and h ◦ f at the fixed point are different. This gives that V is C r -dense.To show that weak unstable foliations of diffeomorphisms from V are non-absolutely continuous we start with some simple observations. First, notice that dueto ergodicity conditional measures cannot have absolutely continuous and singularcomponents simultaneously. Next, it follows from the absolute continuity of W u and the uniqueness of the system of conditional measures of m that the conditionalmeasures of m on the leaves of W wu are equivalent to the conditional measures ofthe induced volume on the leaves of W u . Therefore we only need to look at twodimensional plaques of W u foliated by plaques of W wu . It turns out that absolutecontinuity of W wu inside the leaves of W u is equivalent to W wu being Lipschitzinside W u . Lipschitz property, in turn, can be related to the periodic eigenvaluedata along W su .Pick a plaque P u of W u and let T ⊂ P u and T ⊂ P u be two smooth compacttransversals to W wu with holonomy map p : T → T . If p is Lipschitz for anychoice of plaque and transversals then we say that W wu is Lipshitz inside W u .Theorem A follows from the following lemmas Lemma 3.
Foliation W wu is Lipschitz inside W u if and only if f ∈ U\V . Lemma 4.
Foliation W wu is Lipschitz inside W u if and only if W wu is absolutelycontinuous inside W u . Proofs
Let us begin with a useful observation. If one needs to show that W wu is Lipschitzin a plaque P u then it is sufficient to check Lipschitz property of the holonomy mapfor pairs of transversals that belong to a smooth family that foliates P u , e.g., plaquesof W su . Therefore we can always assume that the transversals are plaques of W su . Proof of Lemma 3.
First assume that f ∈ U\V . Then Lipschitz property of W wu is shown below by a standard argument that uses Livshits Theorem.Let T and T be two local leaves of W su in a plaque P u and let p : T → T bethe holonomy along W wu .For x, y ∈ T with d su ( x, y ) ≥ d su ( p ( x ) , p ( y )) ≤ Cd su ( x, y ) , d su ( x, y ) ≥ , (1)follows from compactness for uniformly bounded plaques P u . It might happen that f n ( x ) and f n ( p ( x )) are far from each other on W wu ( f n ( x )). Hence we need (1)with uniform C not only on plaques P u of bounded size but also on plaques that arelong in the weak unstable direction. In this case (1) cannot be guaranteed solelyby compactness but easily follows from Proposition 1.For x and y close to each other we may use ˜ d su rather than d su since ˜ d su is givenby an integral of a continuous density. Then˜ d su ( p ( x ) , p ( y ))˜ d su ( x, y ) = n − Y i =0 D suf ( f i ( p ( x ))) D suf ( f i ( x )) · ˜ d su ( f n ( p ( x )) , f n ( p ( y )))˜ d su ( f n ( x ) , f n ( y )) , ATHOLOGICAL FOLIATIONS 7 where n is chosen so that d su ( f n − ( x ) , f n − ( y )) < ≤ d su ( f n ( x ) , f n ( y )). The Lip-schitz estimate follows since according to the Livshits Theorem D suf is cohomologousto λ and therefore the product term equals to F ( f n ( x )) F ( f n ( p ( x ))( F ( x ) F ( p ( x ))) − for some positive continuous transfer function F .Now let us take f from V . Specification implies that the closure of the set { λ su ( x ) : x periodic } is an interval [ λ su − , λ su + ]. By applying Anosov Closing Lemmait is possible to change the Riemannian metric so that the constants λ − and λ + from Definition 3 are equal to λ su − / (1 + δ ) and λ su + (1 + δ ) correspondingly. Here δ is an arbitrarily small number.Next we choose periodic points a and b such thatmax (cid:26) λ su + λ su ( a ) , λ su ( b ) λ su − (cid:27) ≤ δ and λ su ( b ) λ su ( a ) ≤ δ ) /γ . This is possible if δ is small enough. From now on δ will be fixed. Constant γ doesnot depend on our choice of a and b and hence δ . It will be introduced later.Denote by n the least common period of a and b . Take ˜ a ∈ W su ( a ) such that d su ( a, ˜ a ) = 1. If one considers an arc of a leaf of W wuL of length D then it is easyto see that this arc is const/ √ D -dense in T . Since conjugacy h f between f and L is H¨older continuous, Proposition 1 implies that an arc of W wu ( a ) of length D is C /D α -dense in T , α >
0. It follows that there exists a point c ∈ W wu ( a )such that d wu ( a, c ) ≤ D , d ( c, b ) ≤ C /D α and W s ( b ) intersects the arc of strongunstable leaf W su ( c ) that connects c and ˜ c = W su ( c ) ∩ W wu (˜ a ) at point ˜ b as shownon the Figure 1. W wu ( a )˜ a W su ( a ) W wu (˜ a ) ˜ cW s ( b ) c ˜ bba Figure 1.
Take N such that d wu ( a, f − n N ( c )) ≤ < d wu ( a, f − n ( N − ( c )). Now our goalis to show that the ratio ˜ d su ( a, f − n N (˜ a ))˜ d su ( f − n N ( c ) , f − n N (˜ c )) ATHOLOGICAL FOLIATIONS 8 can be arbitrarily small which would imply that W su is not Lipschitz. Note thatwe cannot take a smaller N since f − -orbit of c has to come to a local plaque about a . Remark.
We use ˜ d su for convenience. Somewhat messier estimates go through ifone uses d su directly.To estimate the denominator we split the orbit { c, f − ( c ) , . . . f − n N ( c ) } into twosegments of lengths N and N , N + N = n N . Choose N so that d ( f − N ( b ) , f − N ( c ))is still small enough to provide the estimate on the strong unstable derivative D suf ( f − i ( c )) ≤ (1 + δ ) λ su ( b ) , i = 1 , . . . N + 1 . The remaining derivatives will be estimated boldly D suf ( · ) ≤ λ + . Since b and ˜ b are exponentially close — d s ( b, ˜ b ) ≤ C /D α ≤ const · µ − n N − — we seethat there exists β = β ( α, ν − , µ − ) which is independent of N such that N > βN .Proposition 1 implies that the ratio ˜ d su ( a, ˜ a ) / ˜ d su ( c, ˜ c ) is bounded independentlyof D (and N ) by a constant C . We are ready to proceed with the main estimate.˜ d su ( a, f − n N (˜ a ))˜ d su ( f − n N ( c ) , f − n N (˜ c )) = n N +1 Y i =1 D suf ( f i ( c ))( λ su ) n N · ˜ d su ( a, ˜ a )˜ d su ( c, ˜ c ) ≤ ( λ su ( a )) − n N (1 + δ ) N ( λ su ( b )) N (1 + δ ) N ( λ su + ) N C ≤ (1 + δ ) N + N (cid:18) λ su ( b ) λ su ( a ) (cid:19) N (cid:18) λ su + λ su ( a ) (cid:19) N C ≤ (1 + δ ) N +2 N (cid:18) λ su ( b ) λ su ( a ) (cid:19) γ ( N +2 N ) C ≤ (cid:18)
11 + δ (cid:19) n N + N C , where γ = β/β + 2 so that N ≥ γ ( N + 2 N ). The last expression goes to zero as D → ∞ , N → ∞ . Thus W wu is not Lipschitz. (cid:3) Proof of Lemma 4.
Obviously W wu being Lipschitz implies transverse absolute con-tinuity property and hence absolute continuity. We have to establish the otherimplication.Assume that W wu is absolutely continuous in the sense of Definition 4. A priori,conditional densities are only L -functions. Our goal is to show that the densitiesare continuous. Moreover, for m almost every x the density ρ x ( y ) on a plaque P wu satisfies the equation ρ x ( y ) ρ x ( x ) = Y n ≥ D wuf ( f − n ( x )) D wuf ( f − n ( y )) , (2)where D wuf ( z ) = k Df (cid:12)(cid:12) E wuf ( z ) ( z ) k . The expression on the right hand side of theformula is a positive continuous function in y .Consider a full volume set where positive ergodic averages coincide for all con-tinuous functions. By absolute continuity this set should intersect a plaque P wu bya positive leaf-volume m wu set Y . Denote by m Y restriction of m wu to Y . For any y ∈ Y consider measures∆ n ( y ) = 1 n n − X i =0 δ f i ( y ) , µ n = Z Y ∆ n ( y ) dm wu ( y ) . ATHOLOGICAL FOLIATIONS 9
Sequences { ∆ n ( y ) } , y ∈ Y , converge weakly to m . Hence µ n converges to m aswell. Notice that µ n = 1 n n − X i =0 Z Y δ f i ( y ) dm wu ( y ) = 1 n n − X i =0 ( f i ) ∗ ( m Y ) . In case when Y is a plaque of W wu the latter expression is known to converge to ameasure with absolutely continuous conditional densities on W wu that satisfy (2).This was established in [PS83] in the context of u -Gibbs measures, however theproof works equally well for any uniformly expanding foliation such as W wu . Forarbitrary measurable Y same conclusion holds. One needs to use Lebesgue densityargument to reduce the problem to the case when Y is a finite union of plaques. Remark.
The argument presented above can also be found in [BDV05], Section 11.2.2,in the context of u -Gibbs measures.Take an m -typical plaque P u whose boundaries are leaves of W wu and transver-sals T and T as shown on the Figure 2. Then plaque P u is foliated by the plaques P wu ( x ), x ∈ T . As usual, denote by p : T → T the holonomy map. Lipschitzproperty of p will be established by comparing volumes of small rectangles R and R built on corresponding segments of T and T . R T T P u p ( x ) p ( y ) x y R Figure 2.
Denote by µ P u the conditional measure on P u . The conditional densities ρ x ( · ) of m on the plaques P wu ( x ) , x ∈ T , are the same as conditional densities with respectto µ P u .Fix x, y ∈ T and small ε > ε ≪ m T ([ x, y ]). Build rectangles R and R onthe segments [ x, y ] and [ p ( x ) , p ( y )] so that m P wu ( z ) ( R ∩ P wu ( z )) = m P wu ( z ) ( R ∩ P wu ( z )) = ε for every z ∈ [ x, y ]. Then µ P u ( R i ) = Z [ x,y ] d ˆ µ ( z ) Z P wu ( z ) ∩ R i ρ z ( t ) dm P wu ( z ) ( t ) , i = 1 , , where ˆ µ is projection of µ P u to T . These formulae together with uniform con-tinuity of the conditional densities that is guaranteed by (2) imply that the ratio µ P u ( R ) /µ P u ( R ) is bounded away from zero and infinity uniformly in x and y .Since µ P u has positive continuous density with respect to m P u the same conclusion ATHOLOGICAL FOLIATIONS 10 holds for m P u ( R ) /m P u ( R ) and therefore also for m T ([ x, y ]) /m T ([ p ( x ) , p ( y )]). (cid:3) Appendix
Appendix is devoted to the proofs of Propositions 1 and 2. Both proofs rely onsimple growth arguments and a result of Brin, Burago and Ivanov. We will workon the universal cover R and we will indicate this by using tilde sign for liftedobjects. For example, the lift of foliation W suf to R is denoted by f W suf .The result of Brin, Burago and Ivanov [BBI09] says that lifts of leaves of strongunstable foliation are quasi-isometric. Namely, if d is the usual distance then ∃ C > ∀ x, y with y ∈ ˜ W su ( x ) , d su ( x, y ) ≤ Cd ( x, y ) . Proof of Proposition 1.
We argue by contradiction. Assume that ˜ h f ( f W wuf ) = f W wuL .Then we can find points a , b and c with the following properties b ∈ f W wuL ( a ) , c / ∈ f W wuL ( a ) , h − f ( c ) = f W wuf (˜ h − f ( a )) ∩ f W suf (˜ h − f ( b )) . We iterate automorphism L and look at the asymptotic growth of the distancebetween these points. Obviously, distance between images of a and b grows as µ n ,meanwhile distance between images of a and c , and images of b and c grows as λ n .Since conjugacy ˜ h f is C -close to Id we have the same growth rates for the triple˜ h − f ( a ), ˜ h − f ( b ) and ˜ h − f ( c ) as we iterate dynamics ˜ f . Points ˜ h − f ( a ) and ˜ h − f ( c ) lieon the same weak unstable manifold, therefore, constant µ + from the Definition 2is not less than λ . Then, obviously, λ − > λ . Since f W suf is quasi-isometric d ( ˜ f n (˜ h − f ( c )) , ˜ f n (˜ h − f ( b ))) ≈ d su ( ˜ f n (˜ h − f ( c )) , ˜ f n (˜ h − f ( b ))) & λ n − , n → ∞ . On the other hand, we have already established that the distance between imagesof ˜ h − f ( c ) and ˜ h − f ( b ) diverges as λ n . This gives us a contradiction. (cid:3) Proof of Proposition 2.
We argue by contradiction. Assume that f ∈ U\V . Thenfor every periodic point x , λ su ( x ) = ¯ λ = λ .First assume that ¯ λ < λ . Then constant λ + from Definition 2 can be taken tobe equal to ( λ + ¯ λ ). Pick points a and b , b ∈ f W su ( a ). Then d ( ˜ f n ( a ) , ˜ f n ( b )) ≤ d su ( ˜ f n ( a ) , ˜ f n ( b )) . λ n + , n → ∞ . By Proposition 1 ˜ h f ( b ) / ∈ f W wu (˜ h f ( a ). Therefore, d ( ˜ L n (˜ h f ( a )) , ˜ L n (˜ h f ( b ))) & λ n , n → ∞ . On the other hand, d ( ˜ L n (˜ h f ( a )) , ˜ L n (˜ h f ( b ))) = d (˜ h f ( ˜ f n ( a )) , ˜ h f ( ˜ f n ( b ))) . λ n + , n → ∞ , since ˜ h f is C -close to Id . The last two asymptotic inequalities contradict to eachother.Now let us assume that ¯ λ > λ . In this case we can take λ − from Definition 2 tobe equal to ( λ + ¯ λ ). Take a and b as before. Since f W su is quasi-isometric d ( ˜ f n ( a ) , ˜ f n ( b )) & d su ( ˜ f n ( a ) , ˜ f n ( b )) & λ n − , n → ∞ . On the other hand, d ( ˜ f n ( a ) , ˜ f n ( b )) ≈ d (˜ h f ( ˜ f n ( a )) , ˜ h f ( ˜ f n ( b ))) = d ( ˜ L n (˜ h f ( a )) , ˜ L n (˜ h f ( b ))) . λ n , n → ∞ , ATHOLOGICAL FOLIATIONS 11 which gives us a contradiction in this case as well. (cid:3)
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