Robustly N-expansive surface diffeomorphisms

We give sufficient conditions for a diffeomorphism of a compact surface to be robustly $N$-expansive and cw-expansive in the $C^r$-topology. We give examples on the genus two surface showing that they need not to be Anosov diffeomorphisms. The examples are axiom A diffeomorphisms with tangencies at wandering points.


Introduction
Let M be a smooth compact manifold without boundary and consider a C 1diffeomorphism f : M → M . We say that f is expansive if there is a positive constant δ such that if x, y ∈ M and x = y then there is n ∈ Z such that dist(f n (x)f n (y)) > δ, where dist is a metric induced by a Finsler · on the tangent bundle T M . We say that f is C r -robustly expansive if it is in the interior of the set of expansive C r -diffeomorphisms. In [15] Mañé proved that f is C 1 -robustly expansive if and only if it a quasi-Anosov diffeomorphism, i.e., for every tangent vector v ∈ T M , v = 0, the set { df n (v) } n∈Z is unbounded. Also, he proved that f is quasi-Anosov if and only if it satisfies Smale's Axiom A and the quasitransversality condition of stable and unstable manifolds: T x W s (x) ∩ T x W u (x) = 0 for all x ∈ M . If M is a compact surface then every quasi-Anosov diffeomorphism is Anosov. In higher dimensional manifolds there are examples of quasi-Anosov diffeomorphisms not being Anosov, see for example [9]. Obviously, every quasi-Anosov C r -diffeomorphism is C r -robustly expansive. To our best knowledge it is unknown whether the converse is true for r ≥ 2.
The results of [15] were extended in several directions. In [3] Lipschits perturbations of expansive homeomorphisms with respect to a hyperbolic metric were considered. There it is shown that quasi-Anosov diffeomorphisms are robustly expansive even allowing Lipschitz perturbations. In [19] it is shown that a vector field is C 1 -robustly expansive in the sense of Bowen and Walters [7] if and only if it is a quasi-Anosov vector field. In [2] this result is proved for kinematic expansive flows. For vector fields with singular (equilibrium) points Komuro [13] introduced a definition called k * -expansivity. He proved that the Lorenz attractor is k * -expansive, consequently, we have a robustly k * -expansive attractor. The question of determining whether a compact boundaryless three-dimensional manifold admits a C 1 -robustly k * -expansive vector field with singular points seems to be an open problem.
In the discrete-time case the definition of expansivity has several variations. Let us start mentioning the weakest one that will be considered in this paper. We say that f : M → M is a cw-expansive diffeomorphism [11] if there is δ > 0 such that if C ⊂ M is a non-trivial (not a singleton) connected set then there is n ∈ Z such that diam(f n (C)) > δ, where diam(C) = sup x,y∈C dist(x, y). In [22] it is proved that every C 1 -robustly cw-expansive diffeomorphism is quasi-Anosov. In this paper, Section 5, we show that there are C 2 -robustly cw-expansive surface diffeomorphisms that are not quasi-Anosov. For this purpose we introduce the notion Q r -Anosov diffeomorphism. The idea is to control the order of the tangencies of stable and unstable manifolds as will be explained in Section 3.
In [16,17] (see also [18]) Morales introduced other forms of expansivity that will be explained now. The first one saids that f is N -expansive [17], for a positive integer N , if there is δ > 0 such that if diam(f n (A)) < δ for all n ∈ Z and some A ⊂ M then |A| ≤ N , where |A| stands for the cardinality of A. In this case we say that δ is an N -expansivity constant. In [14] examples are given on compact metric spaces showing that N + 1-expansivity does not imply N -expansivity for all N ≥ 1. This extends previous results of [17]. The examples of Section 5 of the present paper are Axiom A diffeomorphisms of a compact surface exhibiting this phenomenon.
From a probabilistic viewpoint expansivity can be defined as follows. For δ > 0, x ∈ M and a diffeomorphism f : M → M consider the set Given a Borel probability measure µ on M we say that f is µ-expansive [16] if there is δ > 0 such that for all x ∈ M it holds that µ(Γ δ (x)) = 0. We say that f is measure-expansive if it is µ-expansive for every non-atomic Borel probability measure µ. Recall that µ is non-atomic if µ({x}) = 0 for all x ∈ M . Also, we say that f is countably-expansive if there is δ > 0 such that for all x ∈ M the set Γ δ (x) is countable. In [4] it is shown that, in the general context of compact metric spaces, countably-expansivity is equivalent to measure-expansivity. In Table 1 we summarize these definitions. The implications indicated by the arrows are easy to prove. Table 1. Hierarchy of some generalizations of expansivity.
As we said before, in [22] it is shown that C 1 -robustly cw-expansive diffeomorphisms are quasi-Anosov. Therefore, in the C 1 -category, all the definitions of Table  1 coincide in the robust sense. The purpose of the present paper is to investigate robust expansivity and its generalizations in the C r -topology for r ≥ 2. Let us explain our results while describing the contents of the article. In Section 2 we recall some known results while introducing the notion of Ω-expansivity, i.e., expansivity in the non-wandering set. We prove that C 1 -robust Ω-expansivity is equivalent with Ω-stability. In Section 3 we introduce Q r -Anosov C r -diffeomorphisms of compact surfaces. In Corollary 3.8 we show that Q r -Anosov diffeomorphisms are C r -robustly r-expansive (i.e., N -expansive with N = r). In Section 4 we investigate the converse of Corollary 3.8. We show that every periodic point of a C r -robustly cw-expansive diffeomorphism on a compact surface is hyperbolic. Also, we prove that if f is an Axiom A diffeomorphism without cycles and C r -robustly cw-expansive then f is Q r -Anosov. In Section 5 we prove that for each r ≥ 2 there is a C r -robustly r-expansive surface diffeomorphism that is not (r − 1)-expansive.

Omega-expansivity
Let M be a smooth compact manifold without boundary. In this section the dimension of M will be assumed to be greater than one. Given a C 1 -diffeomorphism f : M → M define Per(f ) as the set of periodic points of f and the non-wandering set Ω(f ) as the set of those x ∈ M satisfying: for all ε > 0 there is and Ω(f ) is hyperbolic. A compact invariant set Λ ⊂ M is hyperbolic if the tangent bundle over Λ splits as T Λ M = E s ⊕ E u the sum of two sub-bundles invariant by df and there are c > 0 and λ ∈ (0, 1) such that: ∈ Ω(f ), for j = 1, . . . , k, such that α(a j ) ⊂ Λ ij and ω(a j ) ⊂ Λ ij+1 (with k + 1 ≡ 1). We say that f has not cycles (or satisfies the no cycle condition) if there are not cycles among the basic sets of Ω(f ). See for example [21,24] for more on this subject.
From [1,8,10,25] we know that the following statements are equivalent in the C 1 -topology: (1) f satisfies axiom A and has not cycles, We add another equivalent statement, with a simple proof based on deep results, that will be used in the next sections.
Proof. In order to prove the direct part suppose that f is C 1 -robustly Ω-expansive. If a periodic point of f is not hyperbolic then, by [8, Lemma 1.1], we find a small C 1perturbation of f with an arc of periodic points. This contradicts the C 1 -robust expansivity of f because every expansive homeomorphism of a compact metric space has at most a countable set of periodic points. This proves that f is a star diffeomorphism.
If f is Ω-stable then f satisfies Smale's axiom A.
Therefore Ω(f ) is hyperbolic and consequently f :

Q r -Anosov diffeomorphisms
In this section we assume that M = S is a compact surface, i.e., dim(M ) = 2.
. Assume that f is Ω-stable, E s , E u are one-dimensional and define I = [−1, 1]. We denote by Emb r (I, S) the space of C r -embeddings of I in S with the C r -topology. Let us recall the following fundamental result for future reference.
. Finally, these stable manifolds also depend continuously on the diffeomorphisms f , in the sense that nearby diffeomorphisms yield nearby mappings Φ.
such that the graph of g s and g u are the local expressions of the local stable and the local unstable manifold of x, respectively. If the degree r Taylor polynomials of g s and g u at 0 coincide we say that there is an r-tangency at x.
is axiom A, has no cycles and there are no r-tangencies.
Remark 3.3. For r = 1 we have that Q 1 -Anosov is quasi-Anosov, and in fact, given that S is two-dimensional, it is Anosov. For r = 2 we are requiring that if there is a tangency of a stable and an unstable manifold it is a quadratic one.
We will show that Q r -Anosov diffeomorphisms form an open set of N -expansive diffeomorphisms. Several results from [24] will be used. For δ > 0 define , f n (y)) ≤ δ ∀n ≤ 0}. Theorem 3.4. In the C r -topology the set of Q r -Anosov diffeomorphisms of a compact surface is a C r -open set.
Proof. We know that the set of axiom A C r -diffeomorphisms without cycles form an open set U in the C r -topology. Let g k be a sequence in U converging to f ∈ U . Assume that g k is not Q r -Anosov for all k ≥ 0. In order to finish the proof is it sufficient to show that f is not Q r -Anosov. Since g k ∈ U and it is not Q r -Anosov, there is x k ∈ S with an r-tangency for g k .
By [24,Proposition 8.11] we know that Ω(f ) has a local product structure, then, we can apply [24,Proposition 8.22] to conclude that Ω(f ) is uniformly locally maximal, that is, there are neighborhoods U 1 ⊂ S of Ω(f ) and U 2 a C r -neighborhood of f such that Ω(g) = ∩ n∈Z g n (U 1 ) for all g ∈ U 2 . Consider the compact set K = S \ U 1 . We have that for all x / ∈ Ω(g), with g ∈ U 2 , there is n ∈ Z such that g n (x) ∈ K. Notice that if g k has an r-tangency at x k then every point in its orbit by g k has an r-tangency too. Therefore we can assume that x k ∈ K for all k ≥ 1. Since K is compact we can suppose that x k → x ∈ K. By [24, Proposition 9.1] we can take y k , z k ∈ Ω(g k ) such that x k ∈ W s g k (y k ) ∩ W u g k (z k ). Suppose that y k → y and z k → z with y, z ∈ Ω(f ) [24,Theorem 8.3]. By Theorem 3.1 we know that for some δ > 0 the local manifolds W s δ (y k , g k ) and W u δ (z k , g k ) converges in the C r -topology to W s δ (y, f ) and W u δ (z, f ) respectivelly. Since K is compact and disjoint from Ω(f ) there is m > 0 such that x k ∈ g −m k (W s δ (y k , g k )) ∩ g m k (W u δ (z k , g k )) for all k ≥ 1. Then, taking limit k → ∞ we find an r-tangency at x for f . Therefore f is not Q r -Anosov. This proves that the set of Q r -Anosov C r -diffeomorphisms is an open set in the C r -topology. Definition 3.5. We say that a C r -diffeomorphism f is C r -robustly N -expansive if there is a C r -neighborhood of f such that every diffeomorphism in this neighborhood is N -expansive.
The following is an elementary result from Analysis. Lemma 3.6. If g : R → R is a C r functions with r+1 roots in the interval [a, b] ⊂ R then g (n) has r + 1 − n roots in [a, b] for all n = 1, 2, . . . , r where g (n) stands for the n th derivative of g.
Proof. It follows by induction in n using the Rolle's theorem.
Recall that r-expansivity means N -expansivity with N = r.
Theorem 3.7. Every Q r -Anosov diffeomorphism of a compact surface is r-expansive.
Moreover, if f is Q r -Anosov then there are a C r neighborhood U of f and δ > 0 such that δ is an r-expansive constant for every g ∈ U.
Proof. Let f : S → S be a Q r -Anosov diffeomorphism. We can take a neighborhood U of Ω(f ), a C r -neighborhood U of f and δ > 0 such that: (1) Ω(g) is expansive with expansivity constant δ for all g ∈ U and (2) Ω(g) = ∩ n∈Z g n (U ) for all g ∈ U . Therefore, we have to show that there is δ > 0 such that if X ∩ Ω(f ) = ∅ and diam(f n (X)) < δ for all n ∈ Z then |X| ≤ r. Arguing by contradiction assume that there are g n converging to f in the C r -topology and two sequences s n and u n of arcs in S such that s n is stable for g n , u n is unstable for g n , |s n ∩ u n | > r and diam(s n ), diam(u n ) → 0 as n → +∞. Considering the compact set K of the proof of Theorem 3.4 we can assume that s n , u n ⊂ K. Take x ∈ K an accumulation point of s n . By Lemma 3.6 and the arguments in the proof of Theorem 3.4 we have that there is an r-tangency at x for f . This contradiction finishes the proof.
From Theorems 3.4 and 3.7 we deduce: Corolary 3.8. Every Q r -Anosov diffeomorphism of a compact surface is C r -robustly r-expansive with uniform r-expansivity constant on a C r -neighborhood.

Robust cw-expansivity
In this section we will prove that if f is C r -robustly cw-expansive, r ≥ 1, then its periodic points are hyperbolic. It is a first step in the direction of proving the converse of Corollary 3.8 (in case that this converse is true). A second step is done in Theorem 4.4 assuming that the diffeomorphism is Axiom A without cycles.
Moreover, for all r ≥ 0 the function µ → f µ is continuous in the C r -topology.
Proof. Taking a local chart the problem is reduced to Euclidean R n . Then we will assume that M = R n , p = 0 and clos(B s (p)) ⊂ U for some s > 0. Consider a C ∞ function ρ : where · denotes the Euclidean norm. If x < s/2 then f µ (x) = x+(µ−1)x = µx. Therefore, d p f = µI. The rest of the details are direct from (1).
A point x ∈ M is Lyapunov stable for f : M → M if for all ε > 0 there is δ > 0 such that if dist(y, x) < δ then dist(f n (x), f n (y)) < ε for all n ≥ 0. In [12,Theorem 1.6] it is shown that cw-expansive homeomorphisms admits no stable points. This is done for a Peano continuum, as is our compact connected manifold M . This result was previously proved by Lewowicz and Hiraide for expansive homeomorphisms on compact manifolds. This is a key point in the following proof. Arguing by contradiction assume that the eigenvalues of d p f l are smaller or equal than 1 in modulus (being the other case similar). Take from Lemma 4.1 a C r -diffeomorphism f µ of M fixing p and being the identity outside U . In particular, f µ is the identity in a neighborhood of the points f (p), . . . , f l−1 (p). Assume that µ ∈ (0, 1) is close to 1. Define g = f •f µ . In this way p is a periodic point of g of period l, g is C r -close to f and the eigenvalues of d p g l are µλ 1 , . . . , µλ j wich have modulus (strictly) smaller than 1 (being λ 1 , . . . , λ j the eigenvalues of d p f l ). Then p is a hyperbolic sink for g, in particular it is Lyapunov stable. Since f is C r -robustly cw-expansive, we can assume that g is cw-expansive, arriving to a contradiction with [12,Theorem 1.6].
This proposition has the following direct consequence on two-dimensional manifolds: Corolary 4.3. Every C r -robustly cw-expansive diffeomorphism on a compact surface is a C r -star diffeomorphism.
To our best knowledge it is not known whether for r ≥ 2 every C r -star diffeomorphism is Axiom A, even for M a compact surface. The next result is another partial result in the direction of proving the converse of Corollary 3.8.
Proof. We will argue by contradiction assuming that f is not Q r -Anosov. This implies that there is a wandering point p ∈ S with an r-tangency. Take C r local coordinates φ : I × J ⊂ R 2 → S around p, where I, J ⊂ R are open intervals. Since p is a wandering point we can suppose that f n (φ(I × J)) ∩ φ(I × J) = ∅ for all n ∈ Z, n = 0. Let g s , g u : I → J be C r functions such that their graphs describe the local stable and local unstable manifold of p in coordinates. Since there is an r-tangency at p we can suppose that the Taylor polynomials of order r of g s and g u vanishes at 0.
Define the C r diffeomorphism h : I × R → I × R as h(x, y) = (x, g s (x) − g u (x) + y).
Let us show that f µ is not cw-expansive for µ > 0 small. If x 2 + y 2 < µ/2 then j µ (x, y) = h(x, y). Then, we must note that h(x, g u (x)) = (x, g s (x)) and this means that h maps the graph of g u into the graph of g s . For f µ this implies that there is an arc γ = {φ(x, g s (x)) : |x| < µ/ √ 8} ⊂ S such that diam(f n (γ)) → 0 as n → ±∞. Then, arbitrarily small subarcs of γ contradict the cw-expansivity of each f µ for arbitrarily small cw-expansive constants. Now we will show that f µ is a C r small perturbation f if µ is close to 0. By definition, they coincide for q / ∈ φ(B µ (0, 0)). Therefore, the problem is reduced to show that j µ is a C r small perturbation of the identity in I × J. Notice that j µ (x, y)−Id(x, y) = σ( x 2 + y 2 /µ)h(x, y) = (0, g s (x)−g u (x)). In order to conclude we will show that the map is C r -close to (x, y) → (0, 0). Because of the r-tangency at p we know that R(x) = g s (x) − g u (x) satisfies R(x)/x r → 0 as x → 0. This and L'Hospital's rule implies that R (t) (x)/x r−t → 0 as x → 0 for all t = 0, 1, . . . , r. As before, R (t) (x) denotes the t th derivative of R at x. Define ρ(x, y) = σ( x 2 + y 2 ) and let K = ρ C r . Given ε > 0 consider µ > 0 such that for all x ∈ (−µ, µ). For (x, y) ≥ µ we have that ρ(x/µ, y/µ) = 0, so there is nothing to estimate. We will assume that (x, y) ≤ µ. Given two non-negative integers i, j such that i + j ≤ r we have that This proves that f µ is a C r -approximation of f that is not cw-expansive. This contradiction proves the theorem.

Examples of N -expansive diffeomorphisms
In this section we present examples of C r -robustly N -expansive surface diffeomorphisms that are not Anosov. They are variations of the 2-expansive homeomorphism presented in [5], that in turn, is based on the three-dimensional quasi-Anosov diffeomorphism given in [9].
Theorem 5.1. For each r ≥ 2 there is a C r -robustly r-expansive surface diffeomorphism that is not (r − 1)-expansive.
Proof. We start with the case of r = 2. It is essentially the example given in [5], we recall some details from this paper. Consider S 1 and S 2 two copies of the torus R 2 /Z 2 and the C ∞ -diffeomorphisms f i : S i → S i , i = 1, 2, such that: 1) f 1 is a derived-from-Anosov as detailed in [21], 2) f 2 is conjugate to f −1 1 and 3) f i has a fixed point p i , where p 1 is a source and p 2 is a sink. Also assume that there are local charts ϕ i : the pull-back of the stable (unstable) foliation by ϕ 1 (ϕ 2 ) is the vertical (horizontal) foliation on D and Let A be the annulus {x ∈ R 2 : 1/2 ≤ x ≤ 2}. Consider the diffeomorphism ψ : A → A given by ψ( consider the equivalence relation generated by for all x ∈ A. Denote by x the equivalence class of x. The surface ∼ has genus two, we are considering the quotient topology on S. Define the C ∞diffeomorphism f : S → S by We know that f is Axiom A because the non-wandering set consists of a hyperbolic repeller and a hyperbolic attractor. Also f has no cycles. The stable and unstable foliations in the annulus A = ϕ 1 (A) looks like Figure 1. The tangen- cies are quadratic because in local charts stable manifolds are straight lines and unstable manifolds are circle arcs. Then, applying Theorem 3.7 we have that f is C 2 -robustly 2-expansive. It is not 1-expansive (i.e. expansive) because near the line of tangencies we find pairs of points contradicting expansivity (for every expansive constant).
For the case r = 3 we will change ψ in an open set U contained in A. Consider ψ such that the stable and the unstable foliations looks like in Figure 2. There are two curves that are topologically transverse but there is a tangency of order 2. Between these two curves there are points of non topologically transversality. The unstable arcs are modeled by the one parameter family p a (x) = x 3 + (a 2 − 1)x + 9a Figure 2. Dot-lines represents the stable foliation and curved lines are the unstable foliation. In this way f is not 2-expansive but it is C 3 -robustly 3-expansive. for x, a ∈ [−2, 2]. The presence of the term 9a implies that ∂p a /∂a > 0 for all x, a ∈ [−2, 2]. If |a| > 1 then p a (x) > 0 for all x, so we have transversality. If |a| = 1 then p a (x) is a translation of x 3 , so there is a tangency of order 2 at x = 0. If |a| < 1 then p a (x) has a local maximum and local minimum that are close if |a| is close to 1. Therefore, we see that f is not 2-expansive. It is C 3 -robustly 3-expansive because p a (x) = 6 = 0 for all x, a ∈ [−2, 2].
For the case r = 4 we consider an open set U ⊂ A containing a cuadratic tangency as in Figure 3. The map ψ is changed in U in such a way that the unstable arcs corresponds to the curves on the right hand of the figure. They can be modeled with the polynomials p a (x) = x 4 + (a 2 − 1)x 2 + 16a. The general case r ≥ 5 follows the same ideas. For r odd ψ is changed near a box of local product structure. For r even ψ is changed near a quadratic tangency.
In particular we have: Corolary 5.2. There are C r -robustly cw-expansive surface diffeomorphisms that are not Anosov.