Lipschitz perturbations of expansive systems

We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz perturbations with respect to a hyperbolic metric. We also study the relationship between Lipschitz topologies and the $C^1$ topology on smooth manifolds.


Introduction
In the study of dynamical systems with discrete time we can distinguish two important categories: the smooth as for example C 1 diffeomorphisms of smooth manifolds and the topological as homeomorphisms of metric spaces. Here we will consider Lipschitz homeomorphisms, an intermediate one. Hyperbolicity is a key property of smooth systems. In this article we will consider Fathi's hyperbolic metric defined for an arbitrary expansive homeomorphism of a compact metric space to study its Lipschitz perturbations.
Recall that a homeomorphism f : X → X of a compact metric space is expansive if there is δ > 0 such that dist(f n (x), f n (y)) ≤ δ for all n ∈ Z implies x = y. In [2] Fathi proved that for every expansive homeomorphism there are a metric d, defining the original topology, and two parameters λ > 1 and ε > 0 such that if d(x, y) < ε then d(f (x), f (y)) ≥ λ d(x, y) or d(f −1 (x), f −1 (y)) ≥ λ d(x, y). Fathi proposed to called such metric as an adapted hyperbolic metric. The hyperbolic metric is our starting point in the extension of results from hyperbolic diffeomorphisms to expansive homeomorphisms in the Lipschitz category.
A special type of diffeomorphisms with a hyperbolic behavior are the quasi-Anosov diffeomorphisms defined by Mañé. A diffeomorphisms f : M → M of a compact smooth manifold is quasi-Anosov if its tangent map is expansive. Mañé in [10] proved that the interior in the C 1 topology of the set of expansive diffeomorphisms is the set of quasi-Anosov diffeomorphisms. In particular, every C 1 small perturbation of a quasi-Anosov diffeomorphism is expansive, i.e. they are C 1 robustly expansive. In Proposition 3.9 we will show that quasi-Anosov diffeomorphisms are robustly expansive even allowing Lipschitz perturbations.
Let us explain how we arrived to the Lipschitz category. The key is Theorem 6 in Walters paper [17]. There it is proved some kind of structural stability of transitive subshifts of finite type. We will show that Walters perturbations are in fact close in a Lipschitz topology, see Proposition 3.6. In particular he showed that these subshifts are robustly expansive. In this proof he uses the hyperbolicity of the usual distance for the shift map. Using similar ideas, in Corollary 3.8 we conclude that every expansive homeomorphism of a compact metric space is Lipschitz robustly expansive with respect to a hyperbolic metric. Another consequence of our results is that Anosov diffeomorphisms are structurally stable even allowing Lipschitz perturbations, see Corollary 4.7. That is, a homeomorphism obtained after a small Lipschitz perturbation is conjugate to the initial Anosov diffeomorphism. This result extends the one obtained in [16] as explained in Remark 4.8.
Structural stability is related with the shadowing or pseudo-orbit tracing property. See for example [17]. In [13] it is shown that a strong version of the shadowing property (called Lipschitz shadowing) is in fact equivalent with structural stability of C 1 diffeomorphisms. Expansive homeomorphisms of surfaces are known to be conjugate to pseudo-Anosov diffeomorphisms [9,5] and if the surface is not a torus then pseudo-Anosov maps have not the shadowing property. But, as proved by Lewowicz in [8], they are persistent and, in some sense, C 1 structurally stable. See [8] for details and precise definitions. To prove the structural stability Lewowicz assumed that the perturbed homeomorphism coincides with the original one at the singular points of the stable foliation. See [8,Theorem 3.5]. We will show that pseudo-Anosov diffeomorphisms are Lipschitz-structurally stable with respect to a hyperbolic metric, see Corollary 4.9. In our setting we can also prove that singular points cannot be moved with a small Lipschitz perturbation, i.e., Lewowicz's hypothesis holds. This is done in Theorem 4.11.
Let us mention a very special property of the Lipschitz category that is not shared with diffeomorphisms. It is known that there are homeomorphisms of smooth manifolds that are not conjugate to smooth diffeomorphisms. An example can be obtained from the homeomorphism of R 2 given in polar coordinates by f (r, θ) = (r, θ + sin(1/r)).
See [4] and references therein for more on this subject. In the C 2 category a well known example is the Denjoy C 1 circle diffeomorphism (with wandering points) that is not conjugate to a C 2 diffeomorphism. Consequently, from the point of view of topological dynamics, there is some loss of generality in assuming that the dynamic is generated by a C r diffeomorphism. But, every homeomorphisms, even if it is defined on a compact metric space, is conjugate with a bi-Lipschitz homeomorphism. Let us prove it. Given a homeomorphism f : X → X of a compact metric space (X, dist), consider a new distance Now it is easy to see that it defines the original topology and that f and f −1 are Lipschitz with respect to d. The conjugating homeomorphism is id : (X, dist) → (X, d).
See [15] for more on this subject. Therefore, from the viewpoint of topological dynamics, there is no loss of generality in assuming that the homeomorphism is in fact bi-Lipschitz. We wish to present the following question: is every expansive homeomorphism of a compact smooth manifold conjugate to a C 1 diffeomorphism? This seems to be the case in all known examples. This paper is organized as follows. In Section 2 we consider hyperbolic metrics for expansive homeomorphisms and we prove its robust expansiveness allowing perturbations in the topology considered by Walters in [17]. In Section 3 we introduce the Lipschitz topology in the space of bi-Lipschitz homeomorphisms and we prove that it is equivalent with Walters topology. We conclude the Lipschitz robust expansiveness of every expansive homeomorphism with respect to a hyperbolic metric. In particular, quasi-Anosov diffeomorphisms are robustly expansive even allowing Lipschitz perturbations. In Section 4 we apply techniques from [17] to obtain the Lipschitz structural stability (with respect to a hyperbolic metric) of expansive homeomorphisms with the shadowing property. With similar techniques from [8] we obtain the same conclusion assuming that the homeomorphism is persistent instead of having the shadowing property. Applications to Anosov and pseudo-Anosov diffeomorphisms are given. For pseudo-Anosov homeomorphisms we consider a flat Riemannian metric with conical singularities as in, for example, [11,6]. Using this metric we show that singular points of pseudo-Anosov homeomorphisms cannot be moved with a small Lipschitz perturbation. In Section 5 we consider smooth manifolds and we compare Lipschitz topologies with the usual C 1 topology.

Hyperbolic metrics
Let f : X → X be a homeomorphism of a compact metrizable topological space.
In this case δ is an expansive constant and λ is an expanding factor and we also say that f is hyperbolic with respect to dist.
Our first result, Theorem 2.7, is a generalization of [17,Theorem 6]. There it is proved some kind of robust expansiveness of the shift map. We will consider the topology introduced in [17]. Recall that a continuous function f : X → X is Lipschitz if there is k > 0 such that dist(f (x), f (y)) ≤ k dist(x, y) for all x, y ∈ X and a homeomorphisms f is bi-Lipschitz if f and f −1 are Lipschitz.
Remark 2.2. In [2, Theorem 5.1] Fathi proved that every expansive homeomorphism f defined on a compact metric space admits an f -hyperbolic metric defining the original topology of the space. Moreover, f is a bi-Lipschitz homeomorphism with respect to this hyperbolic metric. Definition 2.3. Given two homeomorphisms f, g of X define the C 0 -metric as Given two Lipschitz homeomorphisms f, g : X → X of a compact metric space (X, dist) consider the following pseudometric Remark 2.4. It is not a metric. In fact it holds that dist W (f, g) = 0 if and only if f • g −1 is an isometry of (X, dist). This is a direct consequence of the definition.
If f, g are bi-Lipschitz homeomorphisms define the metric Remark 2.5. Notice that dist W depends on the metric dist of X.
Proof. Let f : X → X be a hyperbolic homeomorphism with respect to the metric dist on X. Let δ > 0 and λ > 1 be an expansive constant and an expanding factor respectively. Take ε > 0 and λ > 1 such that λ − ε > λ . We will show that λ is an expanding factor and δ is an expansive constant for Then λ dist(x, y) ≤ dist(g(x), g(y)). Therefore δ is an expansive constant and λ is an expanding factor for g.
Corollary 2.8. For every expansive homeomorphism f of a compact metrizable space X there is a metric dist in X defining its topology and making f a dist Wrobustly expansive bi-Lipschitz homeomorphism.

Lipschitz topology
The purpose of this section is to prove that dist W induces a Lipschitz topology in the space of bi-Lipschitz homeomorphisms of X.
Definition 3.1. Given two bi-Lipschitz homeomorphisms f, g : X → X of a compact metric space (X, dist) we define and the Lipschitz metric as .
Let us explain why we call it Lipschitz metric.
Definition 3.3. A normed group is a group G with identity e ∈ G and a function · : G → R satisfying: (1) f ≥ 0 for all f ∈ G with equality only at f = e, Every norm induces a distance as Remark 3.4. In some sense, we measure how far are f g −1 and g −1 f from the identity. In [18] it is called as an operational metric.
For example, in the group of homeomorphisms of X we can consider the C 0 -norm And we obtain that dist C 0 of Equation (1) is the distance induced by the C 0 -norm. See for example [1] for more on normed groups. Let L = L(X, dist) denote the group of bi-Lipschitz homeomorphisms of a compact metric space (X, dist). The operation in L is composition. As usual, define the Lipschitz constant of f ∈ L as It is easy to see that: In light of these properties, it is natural to define loglips(f ) = log(lips(f )) and to consider f L = max{loglips(f ), loglips(f −1 )}. We have that (L, · L ) is a semi-normed group. In fact f L = 0 if and only if f is an isometry. In order to obtain a norm we define The following proposition explains why we say that dist L from Equation (3) is a Lipschitz metric in L.
Proposition 3.5. It holds that dist · L = dist L .
Proof. By the definitions we have that Applying the definitions we have: = sup x =y log dist(f (x), f (y)) dist(g(x), g(y)) = dist L (f, g).
And the proof ends.
Proposition 3.6. The metrics dist L and dist W define the same topology on L.
Proof. For f ∈ L define Then, with standard techniques, it can be proved that if dist W (f, g) < δ then dist L (f, g) < kδ. For this purpose, one has to note that The other part of the proof follows by similar arguments.
Corollary 3.8. Every hyperbolic homeomorphism is Lipschitz-robustly expansive. Or equivalently, every expansive homeomorphism of a compact metric space is Lipschitz robustly expansive with respect to an f -hyperbolic metric.
Proof. It is a direct consequence of Theorem 2.7 and Proposition 3.6.
Recall that a C 1 diffeomorphism f : M → M is quasi-Anosov if the set { df n (v) : n ∈ Z} is unbounded for all v = 0. It is known [10] that quasi-Anosov diffeomorphisms are C 1 -robustly expansive (considering C 1 perturbations).
Proposition 3.9. Quasi-Anosov diffeomorphisms are Lipschitz-robustly expansive with respect to every Riemannian metric dist g .
Proof. Let f be a quasi-Anosov diffeomorphism. By Corollary 3.8 it would be sufficient to prove that dist g is hyperbolic for f , but this may not be true because the Riemannian metric may not be an adapted metric. Instead, we will show that dist g is hyperbolic for f n for some n ≥ 1. This is sufficient too. Consider n such that for all Such n exists, as it is proved in [7, Lemma 2.3] (there it is assumed that f is Anosov but it only uses that f is quasi-Anosov). Now, using the exponential map of the Riemannian metric, we can locally lift f to the tangent fiber bundle. In this way we can view f as a small Lipschitz perturbation of df . Consequently, dist g is hyperbolic for f n .

Lipschitz structural stability
In this section we will consider some shadowing properties to study the structural stability allowing Lipschitz perturbations. We say that f ∈ L(X, dist) is Lipschitz structurally stable if for all ε > 0 there is δ > 0 such that if dist L (f, g) < δ then there is a homeomorphism h : X → X such that f h = hg and dist C 0 (h, id) < ε. Definition 4.2. A homeomorphism f : X → X has the weak shadowing property if for all ε > 0 there is δ > 0 such that if dist C 0 (f, g) < δ then for all x ∈ X there is y ∈ X such that dist(f n (y), g n (x)) < ε for all n ∈ Z.
In [12] the expression weak shadowing property is considered with a different meaning. Let us recall that a δ-pseudo-orbit is a sequence x n ∈ X with n ∈ Z such that dist(f (x n ), x n+1 ) < δ for all n ∈ Z. A homeomorphism f : X → X has the (usual) shadowing property if for all ε > 0 there is δ > 0 such that if {x n } n∈Z is a δ-pseudo-orbit then there is y ∈ X such that dist(f n (y), x n ) < ε for all n ∈ Z. This is also called as pseudo-orbit tracing property.
Remark 4.3. The shadowing property implies the weak shadowing property. Theorems 4,5 and 6 in [17] are stated with the shadowing property but the proofs only need the weak shadowing property. To see that this is true, the reader should check that the proofs of Theorems 5 and 6 uses the shadowing property via Theorem 4, and in the proof of Theorem 4 it is used, in its first paragraph, considering a pseudo-orbit that is a true orbit of the perturbed homeomorphism there called as S.
In the general setting of metric spaces, pseudo-orbits may be far from being real orbits of C 0 perturbations as we explain with the following example. there are pseudo-orbits starting at a and finishing at c without real orbits tracing it. In order to prove that f has the weak shadowing property we have to note that every homeomorphism of X, in particular a perturbation of f , has to fix a, b, c and preserve the left part of b in the figure.
Let us give another shadowing property that was introduced in [8].
Note the difference between this definition and the weak shadowing property. It is known that pseudo-Anosov homeomorphisms of surfaces are persistent but they have not the weak shadowing property.
The proof of the following result uses well known techniques that can be found in [17] for the case of the shadowing property.  Proof. Since dist is f -hyperbolic we know by Corollary 3.8 that f is Lipschitz robustly expansive. Therefore, there are σ, α > 0 such that if dist L (f, g) < σ then g is expansive with expansive constant α. In order to prove that f is Lipschitz structurally stable take ε ∈ (0, α/2). Since M is a compact manifold without boundary, we can assume that every continuous function h : In this way we have a function h : M → M . Assume that dist L (f, g) < δ. If h(x) = h(x ) then dist(f n (h(x)), g n (x)) < ε and dist(f n (h(x )), g n (x )) < ε for all n ∈ Z. Therefore, dist(g n (x), g n (x )) < α for all n ∈ Z. Since α is an expansive constant for g we have that x = x . Consequently, h is injective. Let us prove that h is continuous. Assume that x n → x and h(x n ) → y. Taking limit, we have that dist(f n (y), g n (x)) ≤ ε for all n ∈ Z. Also, we know that dist(f n (h(x)), g n (x)) < ε.
Then dist(f n (h(x)), f n (y)) < α for all n ∈ Z. Since α is an expansive constant for f we conclude that y = h(x). Therefore h is continuous. Notice that by (4) we have that dist C 0 (h, id M ) < ε (n = 0). Therefore, h is surjective and a homeomorphism.
In order to show that f • h = h • g fix x ∈ M and note that by (4) it holds that dist(f n+1 (h(x)), g n+1 (x)) = dist(f n (f (h(x))), g n (g(x))) < ε for all n ∈ Z. Applying (4) again, we have that dist(f n (h(g(x))), g n (g(x))) < ε for all n ∈ Z. And by the triangular inequality dist(f n (f (h(x))), f n (h(g(x)))) < α for all n ∈ Z. Then f (h(x)) = g(h(x)) for all x ∈ M because α is an expansive constant of f .
The case of f persistent is similar, using the condition: dist(f n (x), g n (h(x))) < ε for all n ∈ Z, instead of (4).
Notice that in the previous proof we only used that M is a compact manifold, instead of an arbitrary compact metric space, to ensure that small perturbations of the identity are onto. Proof. It is well known that Anosov diffeomorphisms have the shadowing property. If the Riemannian metric is not adapted for f , we consider an adapted one. Note that all Riemannian metrics are Lipschitz equivalent. In this way the Lipschitz topology on bi-Lipschitz homeomorphisms of M does not change and now the Riemannian metric dist g is hyperbolic for f . Then, by Theorem 4.6 the corollary follows.
Remark 4.8. In [16] a result similar to Corollary 4.7 was obtained using a Lipschitz topology defined via local charts. As we will see in Section 5 our Lipschitz topology (induced by dist W ) allows more perturbations. Consequently, Corollary 4.7 is stronger than the result in [16]. Proof. In [8] it is shown that Pseudo-Anosov maps are persistent. Therefore we conclude by Theorem 4.6.
Remark 4.10. For a pseudo-Anosov map f of a compact surface S denote by Sing(f ) the set of singular points of the stable foliation of f . Consider a flat metric in S \ Sing(f ) such that stable and unstable leaves are geodesics. This Riemannian metric induces a f -hyperbolic distance in S. The construction is standard and the details can be found for example in [6, Section 3.1]. An important property of this metric, for our purpose, is that the length of the boundary of a small ball of radius r centered at a singular point p equals nπr, where n is the number of stable prongs of p.
In [8, Theorem 3.5] Lewowicz obtained a result similar to Corollary 4.9 but in the C 1 category. In his hypothesis it is required that the perturbed map coincides with the original one at singular points. In the Lipschitz setting, with a hyperbolic metric, this hypothesis is not needed as the next result shows. Theorem 4.11. Assume that f : S → S is a pseudo-Anosov homeomorphism of a compact surface with a hyperbolic metric as in Remark 4.10. Then there is ε > 0 such that if dist L (f, g) < ε then Sing(g) = Sing(f ).
Proof. If f and g are Lipschitz close then j = f • g −1 and j −1 are Lipschitz close to the identity of S. Then, we can take ε > 0 such that if dist L (f, g) < ε then (5) lips(j) lips(j −1 ) < 3/2.
Let us first show that if dist L (f, g) < ε then j(Sing(f )) = Sing(f ). Assume by contradiction that there is x / ∈ Sing(f ) but j(x) ∈ Sing(f ) with dist L (f, g) < ε. Consider C 1 a small circle of radius r 1 and centered at x such that C 1 ∩ j(C 1 ) = ∅. Let r 2 = min y∈j(C1) dist(j(x), y) and denote by C 2 the circle centered at j(x) with radius r 2 as shown in Figure 2. Since x is a regular point (of the metric) and the metric is flat we have that len(C 1 ) = 2πr 1 , where len denotes the length of the curve. Since j(x) is a singular point we have that (6) len(C 2 ) ≥ 3πr 2 Figure 2. A small Lipschitz perturbation cannot move a singular point.
because a neighborhood of j(x) is constructed by gluing at least 3 flat half-planes as explained in Remark 4.10. We also have that (7) len(j(C 1 )) ≤ lips(j) len(C 1 ) = lips(j)2πr 1 .
Since j = f g −1 and Sing(f ) is f -invariant we have that g(Sing(f )) = Sing(f ). To conclude the proof we have to note that Sing(f ) and Sing(g) are g-invariant finite sets and they are close if ε is small. Therefore, since g is expansive, we conclude that Sing(f ) = Sing(g) as we wanted to prove.

Topologies on the space of smooth functions
In [17, page 238] Walters considered dist W as "analogous to a C 1 metric". In this section we will investigate the relationship between the C 1 topology and two different Lipschitz topologies. In R n we know that every C 1 diffeomorphisms is Lipschitz, therefore, we can restrict a Lipschitz topology to the space of C 1 diffeomorphisms. A natural question is: if two diffeomorphisms are close in a Lipschitz metric are they necessarily C 1 close? This is the problem that will be considered in this section for dist W , defined at (2) above, and the natural Lipschitz metric induced by a vector space structure that will be defined below. In general, the convergence in the C 1 topology implies the convergence in the Lipschitz topology, if we consider smooth manifolds with Riemannian metrics.
For the first result denote by I = [0, 1] the unit interval with the usual metric dist(x, y) = |y − x| for all x, y ∈ I. Denote by Diff 1 (I) the group of C 1 diffeomorphisms of I. The C 1 topology in the one-dimensional case is defined by the distance dist C 0 (f, g) + dist C 0 (f , g ) for f, g ∈ Diff 1 (I). We refer the reader to [14] for the definition of the C 1 topology on general manifolds. Proposition 5.1. In Diff 1 (I) the metric dist W defines the C 1 topology.
Proof. If dist C 0 (f, g) < ε < 1/2 then both diffeomorphisms have derivatives of the same sign. Suppose that f and g are increasing, the other case being similar. Assume that dist W (f, g) ≤ ε. Therefore Fixing x and taking limit as y → x + we have that |f (x) − g (x)| ≤ ε. Since this holds for all x ∈ I we have that f and g are C 1 -close.
Remark 5.2. The argument of the previous proof applied for diffeomorphisms of a compact ball in R n will gives us Let us show that the previous proposition does not hold for higher dimensional manifolds. For ε > 0 given we will construct a diffeomorphism f : D → D, where D = {(x, y) ∈ R 2 : x 2 + y 2 ≤ 1}, so that dist W (f, id) < ε and d (0,0) f is a rotation of angle π. For this purpose we need the following preliminary result. Proof. For k ∈ R consider the function g : R + → R given by g(r) = k − ε log(r).
Notice that g satisfies the differential equation rg (r) = −ε for all r > 0. It is easy to see that if k < 0 and |k| is sufficiently large then there is d ∈ (0, ε) such that g(d) = 0. Fix this value of k and take c ∈ (0, d) so that g(c) = π. Now using bump π 1 c d g ε γ Figure 3. The function γ of Lemma 5.3. In dotted line the function g.
functions near the points c and d it is easy to obtain the function γ. In Figure 3 the construction is illustrated. The numbers a and b are obtained in (0, c) and (d, ε) respectively.
Denote by R θ : R 2 → R 2 the rotation of angle θ around the origin. Notice that f (p) = R π (p) if p ≤ a and f (p) = p if p ≥ b. Also, f (p) = p for all p ∈ D. Therefore, f is C 0 close to the identity of D if ε is small and for all ε the differential d (0,0) f = R π . Consequently f is C 1 far from the identity. We will show that f is close to the identity for the metric dist W . For this, we will first show that the differential of f is close to a rotation at each p ∈ D.
Lemma 5.4. For all p ∈ D it holds that d p f − R γ(r) ≤ ε where r = p .
Proposition 5.5. For the diffeomorphism f it holds that dist W (f, id D ) ≤ ε.
Therefore similar estimates gives us q − p ≤ (1 + ε) f (q) − f (p) Therefore we have proved: Theorem 5.6. There is a sequence of C 1 diffeomorphisms f n of the compact disk D ⊂ R 2 do not converging to the identity in the C 1 topology but satisfying that dist W (f n , id D ) → 0 as n → ∞.
Let us now consider another Lipschitz topology. Suppose that M ⊂ R n is a compact C 1 manifold without boundary. Denote by · the Euclidean norm in R n . x − y .