GENERIC DIFFERENTIABILITY OF THE CONJUGACY FOR HYPERBOLIC MAPS

M. Pollicott and C. Semedo

Warwick University and Manchester University

0. Introduction A natural problem in the classification of dynamical systems is to study the type of conjugacies that may exist between different transformations. If we have a hyperbolic system, then by we expect nearby systems to be conjugate by a . Moreover, this homeomorphism may be H¨older continuous, but typically it will not be differentiable. Let S : X → X and T : X → X be Ck expanding map on the unit circle, with 0 0 k ≥ 2. In particular, we assume that minx∈X |S (x)| and minx∈X |T (x)| > 1. In the classification of such expanding maps one can consider conjugacies π : X → X, i.e., continuous maps for which π ◦ T = S ◦ π. Any two expanding maps of the same degree and orientation are necessarily conjugate. We say that S and T are Cr equivalent if the conjugacy π is Cr. A particularly simple way to construct examples of conjugacies is by structural stability. Given any C1 expanding map f : X → X and a sufficiently close C1 expanding map g : X → X there is a continuous conjugacy map π : X → X (which is close to the identity). We recall the following classic result of Shub and Sullivan [8].

Theorem 1 (Shub-Sullivan). If S and T are Ck expanding maps on the unit circle, with k ≥ 2. Assume that π : X → X is absolutely continuous then S and T are Ck−1 equivalent.

The following claim appears in the work of Ferriera and Pinto [1].

Conjecture (Explosion of Differentiability for circles). Let S : X → X and T : X → X be Ck expanding maps, with k ≥ 2. Let π : X → X be a topological conjugacy. Assume that π is differentiable at a single point then π is C1.

Ferriera and Pinto showed that related properties hold (where differentiability is

Typeset by AMS-TEX 1 2 M. POLLICOTT AND C. SEMEDO

replaced by a weaker (uaa) property). For each x ∈ Y we define the following

f(x + δ) − f(x) D+f(x) = lim sup δ&0 δ f(x) − f(x − δ) D−f(x) = lim sup δ&0 δ f(x + δ) − f(x) D+f(x) = lim inf δ&0 δ f(x) − f(x − δ) D−f(x) = lim inf . δ&0 δ

Definition. Following the standard usage of terminology say that f is differentiable at x ∈ Y if the four limits above are equal. We say that X is differentiable almost everywhere if there exists a Borel set Y ⊂ X with leb(X − Y ) = 0 such that for each x ∈ Y the f is differentiable at x ∈ Y (i.e, the four limits are equal). In contrast to the above conjecture, we have the following result. Theorem 2. Any conjugacy π : X → X is necessarily differentiable at almost every point x ∈ X. However, for generic pairs of maps S and T the associated conjugacy is not C1.

By a generic pairs of maps we mean a dense Gδ set in the space of pairs of such maps in C1(X,X) × C1(X,X) with the usual topology. In particular, Theorem 2 shows that conjugacy can be differentiable on a quite large set (full measure) without necessarily being C1.

1. Proof of Theorem 2 Let S : X → X and T : X → X again denote expanding maps. Assume that they have the same degree d ≥ 2 (i.e., Card(T −1x) = Card(S−1x) = d, for every x ∈ X) and the same orientation (i.e., the derivatives T 0 and S0 have the same sign). For definiteness, we shall assume they preserve the orientation of X (i.e., the sign of the derivative is positive). The other case is similar. Lemma 1.1. Let S : X → X and T : X → X be C1 expanding maps of X with the same orientation and degree. There exists a homeomorphism π : X → X which is a topological conjugacy. Moreover, if S and T are sufficiently close in the C1 topology then the conjugacy will be α-H¨olderfor 0 < α < 1 arbitrarily large. Proof. The proof of the existence of the conjugacy is fairly simple, cf. [4]. Moreover, one can see from the this proof the validity of the second part of the lemma. ¤ The second part of the Lemma is similar to a result of Palis and Viana for Horsehoes, where they showed that for sufficiently small perturbations of horseshoes the conjugacy will have a H¨olderexponent arbitrarily close to unity [6]. We can let X be identified with the unit interval [0, 1] where the end points are identified. We can consider the lift πe : R → R of the map π : X → X with πe(x + 1) = πe(x) + 1. In particular, for the canonical map h : R → X such that h(x + 1) = h(x) we have that h ◦ πe ◦ h−1 = π. Observe that πe is a monotone map. GENERIC DIFFERENTIABILITY OF THE CONJUGACY FOR HYPERBOLIC MAPS 3

Lemma 1.2. Any monotone map f : R → R on R is differentiable at almost every point. If for almost every point x we denote the derivative of f by f 0(x), then

Z b f 0(x)dx ≤ f(b) − f(a), a for all a < b. Proof. This is a classical result. A particularly nice account appears in [7] or in the notes http://www.math.uiuc.edu/∼mjunge/54004-diffmon.pdf ¤ We can apply Lemma 1.1 to the lift πe of the conjugacy π. This allows us to deduce the following. Lemma 1.3. The conjugacy π is differentiable almost everywhere.

Proof. Clearly, the map π is differentiable at x0 ∈ X if and only if πe is differentiable at the lifted point xe0 ∈ R. In particular, we deduce from Lemma 1.1 that π : X → X is differentiable at almost every point of X, as required. ¤ For almost every point x we denote the derivative of π at x by π0(x). Remark. A naive idea would be to try to write

Z 1 1 = πe(1) − πe(0) = (πe)0(u)du (1.1) 0 and deduce that (πe)0(u) 6= 0 on a set of positive measure. This holds for functions π which are absolutely continuous (e.g., when π is C1). However, this identity doesn’t hold without additional hypotheses. For example, it fails for the usual “Devil’s staircase”, which has zero derivative almost everywhere. We can iterate the conjugacy identity π ◦ T = S ◦ π to write

π ◦ T p = Sp ◦ π, for p ≥ 1. (1.2)

Assume now that π : X → X is a C1 conjugacy. By the chain rule we can consider the derivative of (1.2) at almost every point in X to write

(π0 ◦ T p)(T p)0 = ((Sp)0 ◦ π)π0. (1.3)

Let us denote U = {u : π0(u) 6= 0}. The following lemma describes this set. Lemma 1.3. If π0 is continuous then the set U is open and non-empty. Proof. Since π0 is assumed continuous, the set U is open and nonempty, since we are assuming π0 is continuous and so (1.1) does now hold. ¤ Lemma 1.4. The periodic points are dense in X. Proof. This is a classical result, cf. [4]. It is easily seen using a Markov partition, say. ¤ We can now prove the second part of Theorem 2. Assume that π is C1 then by Lemma 1.3 the set U is open and nonempty. In particular, by Lemma 1.1 we 4 M. POLLICOTT AND C. SEMEDO

p can choose a x0 ∈ U. Assume that S (x0) = x0, then the image p y0 = π(x0) satisfies T (y0) = y0. We can now evaluate (1.3) at x0, to write

0 p 0 p 0 0 π (x0)(T ) (x0) = (S ) (y0)π (y0). (1.4)

0 Let p be the period of x0. Since, by hypothesis, π (x0) 6= 0 we now see from (1.4) that either p 0 p 0 0 (1) (T ) (x0) = (S ) (y0) and π (y0) 6= 0 or p 0 0 (2) (T ) (x0) = 0 and π (y0) = 0 p 0 However, by hypothesis |(T ) (x0)| > 1, so the second possibility can never occur. Moreover, generically the first possibility won’t hold either. More precisely, let

p 0 p 0 Ux0 = {(S, T ):(T ) (x0) 6= (S ) (y0)}

then this set is open and dense sets in space of pairs of orientation preserving degree d maps in C1(X,X) × C1(X,X). We denote by

∞ n U = ∩n=1 ∩x0∈Fix(T ) Ux0

the intersection of this countable family of open dense sets. The result then follows by the Baire Category Theorem. ¤

2. Surface diffeomoprhisms We can also consider a two dimensional analogue of Theorem 2. In particular, we shall concentrate on Anosov surface diffeomorphisms. Let f : M → M be a C2 diffeomorphism on a compact manifold. We say that f is Anosov if there is a continuous Df-invariant splitting of the tangent space TM = Es ⊕ Eu and constants C > 0 and 0 < λ < 1 such that

n n s ||Dxf (v)|| ≤ Cλ ||v|| for v ∈ E −n n u ||Dxf (v)|| ≤ Cλ ||v|| for v ∈ E

for all n ≥ 0. We shall also assume that f is transitive (i.e., there exists a dense ). We recall the following beautiful result of de la Llave [5]. Theorem 3 (de la Llave). Let S and T be Ck Anosov maps on a two dimensional surface, with k ≥ 2. Assume that π : X → X is absolutely continuous with respect to the SRB measures then S and T are Ck−1 equivalent. We refer the reader to citeBo for the definition of SRB measures, since we shall not require this result in the sequel. The following claim appears in the work of Ferriera and Pinto [2]: Conjecture (Explosion of Differentiability for surfaces). Let S : X → X and T : X → X be Ck Anosov diffeomorphisms of a compact surface X with k ≥ 2. Let π : X → X be a topological conjugacy. Assume that π is differentiable at a single point then π is C1. GENERIC DIFFERENTIABILITY OF THE CONJUGACY FOR HYPERBOLIC MAPS 5

2 At each point x ∈ X we can choose a natural basis e1, e2 ∈ R for the tangent 2 space TxM (identified with R ). We say that f is differentiable at a point if the partial derivatives ∂f(x) f(x + ²e ) − f(x) = lim 1 ∂x1 |²|→0 ² and ∂f(x) f(x + ²e ) − f(x) = lim 2 ∂x2 |²|→0 ² exist. This is weaker than the Frechet derivative existing at x. We say that f is differentiable almost everywhere if there exists a Borel set Y ⊂ X with leb(X − Y ) = 0 such that for each x ∈ Y these limits exist. In contrast to the above conjecture, we have the following negative answer. Theorem 4. Every conjugacy π : X → X is necessarily differentiable at almost every point x ∈ X. However, generic diffeomorphisms are not C1. For Anosov diffeomorphisms we can use Markov partitions to reduce the question to essentially that of one dimensional expanding maps. Given x ∈ M we denote

W +(x) = {y ∈ M : d(f nx, f ny) → 0 as n → +∞}

and W −(x) = {y ∈ M : d(f −nx, f −ny) → 0 as n → +∞}. Let F + = {W +(x)} and F − = {W −(x)} be the stable and unstable foliations for M. The following result is due to Hirsch, Pugh and Shub [3]. Lemma 2.1. The leaves W +(x) and W −(x) are C2 immersed one dimensional manifolds. The foliations F + and F − are C1+α, for some α > 0. In particular, for every x ∈ M we can find a neighbourhood U of x and a C1+α diffeomorphism ψ : U → R2 which maps the leaves of F + restricted to U on the horizontal lines in R2. Similarly, for every x ∈ M we can find a neighbourhood V of x and a C1+(α diffeomorphism ψ0 : V → R2 which maps the leaves of F − restricted to U on the vertical lines in R2. (As usual, C1+α means that the maps are C1 and that the derivatives are α-H¨oldercontinuous). Let ² > 0 be sufficiently small. If x, y ∈ M are close enough then [x, y] := + − W² (x) ∩ W² (x) is a single point where

+ n n W² (x) = {y ∈ M : d(f x, f y) ≤ ² for n ≥ 0)} and − −n −n W² (x) = {y ∈ M : d(f x, f y) ≤ ² for n ≥ 0)}.

We can consider a Markov partition T = {T1,...,Tk} for the Anosov diffeo- morphism. Each element can be written in the form Ti = [Ui,Si], for i = 1, . . . , k. where Ui and Si are pieces of unstable and stable manifolds. These are closed sets such that k (1) M = ∪i=1Ti; (2) Ti = int(Ti), for 1 ≤ i ≤ k; (3) if x ∈ int(Ti) and fx ∈ int(Tj) then f([x, Si]) ⊂ [T x, Sj]; and −1 −1 (4) if x ∈ int(Ti) and f x ∈ int(Tl) then f ([Ul, x]) ⊂ [Ul, x]. 6 M. POLLICOTT AND C. SEMEDO

By considering the natural projections along the stable and unstable foliations, + + + + k the map f induces a Markov expanding map f : Xf → Xf , where Xf := ∪i=1Ui; −1 − − − − and the map f induces a Markov expanding map f : Xf → Xf , where Xf = k ∪i=1Si. 1 + + + Lemma 2.2. If the map f : M → M is C then both T : Xf → Xf and − − − 1 T : Xf → Xf are C (on points interior to their intervals of definition). Proof. This follows easily using the C1 property of foliations F + and F − to identify Ti with Si × Ui. ¤ Assume that π : M → M is a conjugacy between two Anosov diffeomorphisms f : M → M and g : M → M. We can define Markov partitions for g by S = {S1,...,Sk}. Lemma 2.3. The map π maps the stable (unstable) foliation for f to the stable (unstable) foliation for g. Moreover, the map π maps the Markov Partition T = {T1, ··· ,Tk} for f to the Markov Partition S = {S1, ··· ,Sk} for g, where Si = π(Si) for 1 ≤ i ≤ k. Proof. This follows from the definition of the stable and unstable manifolds and the conjugacy identity. ¤ The Markov partitions also allow us to relate the pairs of one dimensional ex- panding maps for each Anosov diffeomomorphism by a pair of one dimensional conjugacies. The conjugating map π : M → M induces a natural conjugacy π+ : + + + + + + + + + + + + Xf → Xg between f : Xf → Xf and g : Xg → Xg , (i.e., π f = g π ). − − − Similarly, π : M → M induces a natural conjugacy π : Xf → Xg between − − − − − − − − − − f : Xf → Xf and g : Xg → Xg , (i.e., π f = g π ). We can now proceed with the proof of Theorem 4. Given π : M → M we can associate the two conjugacies π+ and π−. By applying Theorem 2 we can deduce that both maps are differentiable almost everywhere. Lemma 2.4. Let x be in the interior of one of the elements of the Markov Parti- + tion. The map π : M → M is differentiable at x ∈ intTi if π is differentiable at − the corresponding point x1 ∈ Ui and π is differentiable at the corresponding point x2 ∈ Si. Proof. This is again an easy consequence of the C1 property of the foliations. ¤ In particular, we can deduce that π is differentiable almost everywhere, as re- quired for the first part of Theorem 4. We can now prove the second part of Theorem 4. Let us denote U = {u : det(π0)(u) 6= 0}. This set is non empty and open. In particular, by Lemma 1.1 we can choose a periodic point x0 ∈ U. Let p be the 0 period of x0. Since, by hypothesis, π (x0) 6= 0 we now see from the chain rule applied to π ◦ T p = Sp ◦ π that

p p Dπ(x0)DT (x0) = DS (y0)Dπ(y0), (2.1)

where y0 = π(x0). Thus either p 0 p 0 0 (1) det((T ) )(x0) = det((S ) )(y0) and det(π )(y0) 6= 0 or p 0 0 (2) det((T ) )(x0) = 0 and det(π )(y0) = 0 GENERIC DIFFERENTIABILITY OF THE CONJUGACY FOR HYPERBOLIC MAPS 7

p p 0 Since T is a diffeomorphism we have that det((T ) )(x0) 6= 0, so the second possibility can never occur. Moreover, generically the first possibility won’t hold either. More precisely, let

p 0 p 0 Ux0 = {(S, T ) : det((T ) )(x0) 6= det((S ) )(y0)}

then this set is open and dense sets in space of pairs of orientation preserving degree d maps in C1(M,M) × C1(M,M). We denote by

∞ n U = ∩n=1 ∩x0∈Fix(T ) Ux0

the intersection of this countable family of open dense sets. The result then follows by the Baire Category Theorem. ¤

3. Three dimensional Flows We can also show analogous results for transitive Anosov flows on three dimen- sional manifolds. ∞ Let φt : M → M be a C flow on a d-dimensional compact manifold. We say that φ is a transitive Anosov flow if (1) There exists a splitting TM = E0 ⊕ Es ⊕ Eu and constants C > 0 and λ > 0, such that for t ≥ 0,

−λt s ||Dφtv|| ≤ C||v||e if v ∈ E −λt u ||Dφ−tv|| ≤ C||v||e if v ∈ E ;

(2) There exists a dense orbit for φ.

Let φt : X → X and ψt : Y → Y be two hyperbolic flows. A map π : X → Y is a topological conjugacy if φtπ = πψt for all t ∈ R We recall the following interesting result of de la Llave [5]. Theorem 5 (de la Llave). Let S and T be Ck Anosov flows on a three dimen- sional surface, with k ≥ 2. Assume that π : X → X is absolutely continuous with respect to the SRB measures then S and T are Ck−1 equivalent. The analogue of Theorem 4 is the following. Theorem 6. Every conjugacy π : X → X is necessarily differentiable at almost every point x ∈ X. However, generic diffeomorphisms are not C1. The method of proof is very similar to that in the case of surface diffeomorphisms. In this case we can choose two dimensional Markov sections R = {R1, ··· ,Rk} transverse to the flow direction. In this case the associated Poincar´emap f : k k ∪i=1Ri → ∪i=1Ri can be treated in the same way as the Anosov diffeomorphism in the previous section. In particular, by projecting onto the expanding and con- tracting directions of the sections one can again associate pairs of expanding one dimensional maps. In this case, one can show that given two topologically conju- gate Anosov flows on three dimensional manifolds the conjugacy restricted to the sections is differentiable almost everywhere. (In fact, this conclusion only requires that the flows be flow equivalent, i.e., conjugate up to change of parameterization of the flow). Finally, the conjugacy is automatically differentiable in the flow direction 8 M. POLLICOTT AND C. SEMEDO because it is an isometry. Thus we can conclude that the conjugacy is differentiable almost everywhere at points x lying on sections. However, since the sections are arbitrarily chosen the result follows. Remark. If we assume that the two Anosov flows are only flow equivalent then the same conclusion holds. In this case the flow equivalence doesn’t necessarily preserve the flow parameterization, but it is monotone when restricted to orbits and thus by Lemma 1.2 we can still deduce that the equivalence is differentiable almost everywhere in the flow direction. This applies, for example if we perturb a given Anosov flow to get a second Anosov flow using structural stability.

References 1. F. Ferreira and A. Pinto, Explosion of smoothness from a point to everywhere for conjugacies between Markov families, Dynamical Systems 16 (2001), 193-212. 2. F. Ferreira and A. Pinto, Explosion of smoothness from a point to everywhere for conjugacies between diffeomorphisms on surfaces, Ergod. Th. and Dyn. Sys. 23 (2003), 509-518. 3. M. Hirsch, C. Pugh and M. Shub, Invariant Manifold, Lecture Notes in 583, Springer, Berlin. 4. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, C.U.P., Cambridge, 1995. 5. R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys. 150 (1992), 289-320. 6. J. Palis and M. Viana, Continuity of Hausdorff dimension and limit capacity for horseshoes, Lecture Notes in Math. 1331 (1988), 150-160. 7. C. Pugh, Real mathematical analysis, Undergraduate Texts in Mathematics, Springer, New York, 2002. 8. M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Erg. Th. and Dynam. Sys. 5 (1985), 285–289.

Mark Pollicott, Department of Mathematics, Warwick University, Coventry, CV4 7AL, UK E-mail address: mpollic@ maths.warwick.ac.uk

Carlos Semedo, Department of Mathematics, Manchester University, Oxford Road, Manchester M13 9PL, UK