Holder stability of diffeomorphisms

We prove that a $C^2$ diffeomorphism $f$ of a compact manifold $M$ satisfies Axiom A and the strong transversality condition if and only if it is H\"{o}lder stable, that is, any $C^1$ diffeomorphism $g$ of $M$ sufficiently $C^1$ close to $f$ is conjugate to $f$ by a homeomorphism which is H\"{o}lder on the whole manifold.


Introduction
Let M be a compact C ∞ manifold, Diff 1 (M ) be the group of C 1 diffeomorphisms of M . f ∈ Diff 1 (M ) is structurally stable if for any g ∈ Diff 1 (M ) sufficiently C 1 close to f , there is a homeomorphism h of M such that g = hf h −1 . Recall that f satisfies Axiom A if the nonwandering set Ω of f is hyperbolic and the set of periodic points of f is dense in Ω, f satisfies the strong transversality condition if for any two points x, y ∈ Ω the stable manifold W s (x) intersects the unstable manifold W u (y) transversally. By the Structural Stability Theorem of Robbin, Robinson, Liao and Mañé [8,9,6,7], f ∈ Diff 1 (M ) is structurally stable if and only if f satisfies Axiom A and the strong transversality condition. It is also known that in this case the conjugacy h can be chosen to be Hölder on the nonwandering set Ω of f (see [5,Theorem 19.1.2]).
In this paper, we prove that in the above case, the conjugacy h can be chosen to be Hölder not only on Ω but also on the whole manifold M . We say that a deffeomorphism f of M is Hölder stable if for any g ∈ Diff 1 (M ) sufficiently C 1 close to f , there is a Hölder homeomorphism h of M such that g = hf h −1 (This notion should not be confused with the notion of C r structural stability of a C r diffeomorphism, for which g is C r close to f and the conjugacy h is only required to be continuous). We prove that Axiom A plus the strong transversality condition is also equivalent to Hölder stability. For simplicity, we assume that f is C 2 . Since Hölder stability implies structural stability, to prove Theorem 1.1, it is sufficient by the Structural Stability Theorem to prove that Axiom A plus the strong transversality condition implies Hölder stability.
To state the quantitative result, we recall the notion of hyperbolicity. The nonwandering set Ω of a diffeomorphism f is hyperbolic if the restriction T M | Ω of the tangent bundle T M on Ω admits a T f -invariant continuous splitting T M | Ω = E u ⊕ E s such that for some λ ∈ (0, 1), Here the norm is evaluated with respect to some adapted smooth Riemannian metric on M .
Theorem 1.2. Let f be a C 2 diffeomorphism of a compact C ∞ manifold M satisfying Axiom A and the strong transversality condition. Let λ ∈ (0, 1) be as in (1.1), l = max{Lip(f ), Lip(f −1 )}. Suppose α ∈ (0, 1) satisfies λl α < 1. Then for any C α neighborhood V of the identity map in C α (M, M ), there exists a C 1 neighborhood N of f in Diff 1 (M ) such that for every g ∈ N , there is a homeomorphism h of M in V such that g = hf h −1 , and the assignment g → h is C 1 as a map N → C 0 (M, M ) and sends f to the identity.
Here Lip(f ) denotes the Lipschitz constant of f , C α (M, M ) and C 0 (M, M ) are the Banach manifolds of C α and C 0 maps on M , respectively.
Hölder stabillity over hyperbolic sets is well known ( [5,  One can not expect more regularity of the conjugacy h than to be Hölder. For example, Lipschitz conjugacies almost never exist. But for dynamical systems of large group actions, C r or C ∞ conjugacies may exist (see [1] and the references therein).
Our proof of Theorem 1.2 follows the approach of Robbin-Robinson [8,9], where the result that Axiom A plus the strong transversality condition implies structural stability is proved. As in Robbin [8], we divide the proof into three steps, which are the contents of the following three sections.
In Section 2, we prove that for each component Ω i in the spectral decomposition of Ω, the splitting The proof follows ideas in [8,9]. But since we require that the extended splitting to be Hölder, and the metric d on M , unlike Robbin's metric d f [8], is not f -preserving, we need more careful topological arguments. Indeed, we can only prove that the extended bundles E u i and E s i are Hölder on N n=−N f n (U i ) for every N > 0. But this is sufficient for us to derive further results. In Section 2 we only need the weaker restriction λ 2 l α < 1 on the Hölder exponent α comparing with Theorem 1.2, and the case of α = 1 is allowed, which means as usual that the subbundles are Lipschitz.
Using the extended splitting in Section 2, we prove in Section 3 that the induced operator f ♯ of f on the Banach space of C 0 vector fields has a right inverse which restricts to a continuous linear operator on the Banach space of C α and d f -Lipschitz vector fields. The proof is also motivated by [8]. But as in Section 2, since f does not preserve the metric d, some different topological arguments are needed. The condition of α = 1 is not explicitly used in the proof. But since it is easy to see that l ≥ λ −1 , the inequality λl α < 1 for α = 1 never holds. So the case of α = 1 is automatically excluded.
In Section 4 we finish the proof of Theorem 1.2. We first prove a version of Implicit Function Theorem for Banach spaces involving non-closed subspaces. Then using the result in Section 3, we can apply the Implicit Function Theorem to the C 1 map Ψ : Diff 1 (M ) × C 0 (M, M ) → C 0 (M, M ), Ψ(g, h) = ghf −1 to obtain a fixed point h of Ψ(g, ·) for g sufficiently C 1 close to f , and h is sufficiently C α and d f -Lipschitz close to the identity. As in [8,9], the fact that h is d f -Lipschitz close to the identity implies h is a homeomorphism.
Most arguments concerning C 0 estimates in this paper are borrowed from [8,9] except for a few changes of details. But to introduce notations in order to perform the C α estimates, it seems necessary to repeat some of them.
The author would like to thank Professors Boris Hasselblatt and Lan Wen for useful comments.

Extensions of the splitting
In this section we prove that the splitting T M | Ω = E u ⊕ E s can be extended to a neighborhood of each component of Ω and satisfies certain compatibility condition. This is motivated by [8,Theorem 8 We first collect some standard facts that are used in the proof of Theorem 2.1 below. Most of them can be found in [4,8,10]. Let f be a diffeomorphism of a compact manifold M satisfying Axiom A and the strong transversality condition.
Let Ω = Ω 1 ∪· · ·∪Ω k be the spectral decomposition of the nonwandering set Ω of f . Each Ω i is a closed topological transitive hyperbolic f -invariant subset of M , and E u , E s have constant ranks on Ω i . The components Ω i can be ordered in such a way that Then each Ω i has arbitrarily small unrevisited open neighborhood.
We fix an Ω i . For x ∈ Ω i and δ > 0, let W u δ (x) and W s δ (x) be the local unstable and stable manifolds of size δ at x. Let W σ δ (Ω i ) = x∈Ωi W σ δ (x), σ = u, s. As in [8,9], we introduce the metric d f on M by d f (x, y) = sup n∈Z d(f n (x), d n (y)), where d is the metric induced from some Riemannian metric on M .
Theorem 2.1. Let f be a C 2 diffeomorphism of M satisfying Axiom A and the strong transversality condition, Ω = Ω 1 ∪ · · · ∪ Ω k be the spectral decomposition, and the components Ω i are ordered as above. Let λ ∈ (0, 1) be as in (1.1), l = max{Lip(f ), Lip(f −1 )}. Suppose α ∈ (0, 1] satisfies λ 2 l α < 1. Then for any λ ′ ∈ (λ, 1), there exist for each Proof. We extend the definition of the bundles E u and E s on Ω as E u = {v ∈ T M : We prove this by induction on i = 1, · · · , k. Let 1 ≤ i ≤ k. Suppose that for j < i, U j and E u j have been defined and satisfy (i')-(iv') (for i = 1 nothing is defined). We construct U i and E u i satisfying (i')-(iv').
. Similar to the arguments in [8, page 488-491], we can prove (after possibly shrinking of U j , j < i in the induction hypothesis) that there exist an open neighborhood . By making Q 1 smaller, we may assume that τ 0 is bounded, say τ 0 ≤ r 2 for some r ≥ 1.
Consider the smooth vector bundle L over V 2 whose fiber L For the convenience of the following discussion, we embed M isometrically into some Euclidian space R N . Then for x ∈ V 2 ,Ē s x andĒ u x can be viewed as subspaces of R N , and we have the identification Then for g 1 , g 2 ∈ L with different base points, the summation g 1 + g 2 and its norm g 1 + g 2 make sense, as they are viewed as elements in L(R N , R N ). Let Γ x = Γ| Lx be the restriction of Γ on the fiber L x . Since the map Γ : is Lipschitz and C α , which means that there exists C > 0 such that for any x, y ∈ V 2 ∩ f −1 (V 2 ) and g 1 ∈ L(r) x , g 2 ∈ L(r) y . Note that since Γ covers f and f is Lipschitz, we have indeed omitted a term d(f (x), f (y)) in the left hand side of (2.3).
To simplify notations, we denote Σ is a closed subset of the Banach space of continuous bounded sections of L| V3 . By taking a bump function on M which is 1 in Q 3 and 0 outside Q 2 and enlarging K if necessary, it is easy to see that Σ is nonempty. Define the graph transform F ♯ (τ ) of τ ∈ Σ as the section of L| V3 . We prove that F ♯ maps Σ into Σ and is a contraction on Σ. y) for τ ∈ Σ and x, y ∈ V 3 . There are three cases. ( Suppose d(x, y) < d 0 . Since x ∈ Q 3 , we have y ∈ Q 2 and y / ∈ W u (Ω i ). So there exists n ≥ 1 such that f −n (y) ∈ Q 3 . But Q 2 is unrevisited and Q 2 ∩f 2 (Q 2 ) = ∅. So we must have n = 1 and then F ♯ (τ )(y) = Γ(τ (f −1 (y))) = Γ(τ 0 (f −1 (y))) = τ 0 (y). Similarly, This proves that F ♯ maps Σ into Σ. By (2.2), F ♯ is a contraction on Σ. So there is a fixed pointτ of F ♯ in Σ. Choose Now consider the C α and d f -Lipschitz subbundle T f n (E u i1 ) of T M | f n (V3) , n ∈ Z. If for n, m ∈ Z, n < m, f n ( ). The proof of Theorem 2.1 is finished.

Existence of right inverses
Let f be a C 2 diffeomorphism of a compact manifold M , α ∈ (0, 1). Let X 0 (M ) denote the Banach space of continuous vector fields on M with the C 0 norm · , and let X α f (M ) be the subspace of X 0 (M ) consisting of C α and d f -Lipschitz vector fields. As in the previous section, suppose M is isometrically embedded into some Euclidian space R N . For η ∈ X α f (M ), denote Then X α f (M ), being endowed with the norm ).  Proof. Choose λ < λ ′ < ρ = κλ ′ < 1 such that ρl α < 1. Let Ω = Ω 1 ∪ · · · ∪ Ω k be the spectral decomposition ordered as in Theorem 2.1.
Shrinking U i if necessary, we may assume that they are unrevisited. Then it is easy to see that for every x ∈ M , the set {n ∈ Z : f n (x) / ∈ k i=1 U i } contains at most n 0 = 2kN elements. Let θ 1 , · · · , θ k be a smooth partition of unity subordinate to the above cover. For η ∈ X 0 (M ), let [8] proved that these series converge uniformly, and then J is a continuous right inverse of 1 − f ♯ . We prove in the following that J maps X α f (M ) into X α f (M ) and restricts to a continuous linear operator on X α f (M ). As in [8], it is sufficient to prove this property for each J is .
Let η ∈ X α f (M ). Fix i = 1, · · · , k, and denote ζ for all x ∈ M and n ≥ 0 (note that we always have T f > ρ). Hence is the Hölder constant of P E s j | Uj as a map x → P (E s j )x for x ∈ U j . We prove that for all n ≥ 0, where C ′ = CK 2 l α ρ(l α −1) . We first prove some inequalities on individual tangent vectors. Let p, q ∈ M , v p ∈ T p M, v q ∈ T q M . Then Recall that a smooth adapted Riemannian metric on M can be obtained by approximating a C 0 adapted metric for which the bundles E u and E s are mutually orthogonal on Ω. So after choosing a better approximation of the C 0 metric and shrinking the U j 's, we may assume that for each j, P (E s j )p ≤ κ for every p ∈ U j , where κ > 1 is as in the beginning of the proof. So for p, q, v p , v q as above, if moreover we have p, q ∈ U j for some j, and v p ∈ (E s j ) p , v q ∈ (E s j ) q , then Now we prove (3.3). Let x, y ∈ M, n ≥ 0. If one of f −n (x) and f −n (y) does not belong to (3.4) and using (3.2), we get Then by (3.6) and (3.7), we have k j=i U j } consists of at most n 0 elements. Then for all but at most 2n 0 + N integers m in {1, · · · , n}, f −m (x), f −m (y) ∈ U j for some j ≥ i, that is, at most 2n 0 + N numbers This proves (3.3).
Similarly, we can prove that for some constant A, B > 0 (see [8,Section 6]). Since the bundles E s i and E u i are C α and d f -Lipschitz on N n=−N f n (U i ), the operator on X α f (M ) which maps η to η is is continuous. So by (3.2), (3.9) and (3.10), the operator J is maps X α f (M ) into X α f (M ) and is continuous on X α f (M ). This proves the theorem.

Proof of Theorem 1.2
In this section we prove Theorem 1.2. As indicated in the introduction, Theorem 1.1 follows from Theorem 1.2.
We first extract some analytical arguments in [8,9] to the following lemma, which can be viewed as a generalization of the usual Implicit Function Theorem for Banach spaces.
Lemma 4.1. Let (X, · ) be a Banach space, X ′ be a linear subspace of X with a complete norm · ′ such that the inclusion (X ′ , · ′ ) ֒→ (X, · ) is continuous, and such that the closed unit ball {x ∈ X ′ : x ′ ≤ 1} in X ′ is a closed subset of (X, · ). Let M be a Banach manifold, f ∈ M, U be an open set in X containing 0 ∈ X. Let Ψ : M × U → X be a C 1 map satisfying Ψ(f, 0) = 0 and Ψ(M × (U ∩ X ′ )) ⊂ X ′ . Denote by A = D 2 Ψ(f, 0) : X → X the partial derivative of Ψ at the point (f, 0) along the second variable. Suppose (1) A(X ′ ) ⊂ X ′ ; (2) 1 − A has a continuous linear right inverse J which maps X ′ into X ′ and restricts to a continuous linear operator on X ′ ; (3) for any ε > 0, there exist a neighborhood M ε of f in M and a neighborhood U ε of 0 in U such that Then for any neighborhood V ⊂ X ′ of 0 in (X ′ , · ′ ), there exist a neighborhood N of f in M and a map c : N → V such that (i) c(f ) = 0; (ii) Ψ(g, c(g)) = c(g) for all g ∈ N ; (iii) as a map N → X, c is C 1 .
Proof. Denote the norm of J as a operator on X by J , and the norm of J| X ′ as a operator on X ′ by J ′ . Choose 0 < ε ≤ 1 such that the closed ball B ′ (ε) = {x ∈ X ′ : x ′ ≤ ε} lies in V. By the condition (3) and the continuous differentiability of Ψ, we may choose an open neighborhood N of f in M and r > 0 such that the closed ball B(r) = {x ∈ X : x ≤ r} lies in U, and such that for all g ∈ N , x ∈ B(r) ∩ X ′ . By making N smaller, we may also assume that For g ∈ N , define a map R g : B(r) → X by R g (x) = J(Ψ(g, x) − A(x)).
Then for x ∈ B(r), by (4.1), (4.3) and the Mean Value Theorem, we have ≤r.
So R g maps B(r) into B(r). For x, y ∈ B(r), also by (4.1) and the Mean Value Theorem, we have So R g is a contraction on B(r). By the Contraction Principle, there is a unique fixed point c(g) of R g in B(r). This means that (1 − A)(c(g)) = (1 − A)(R g (c(g))) = Ψ(g, c(g)) − A(c(g)). So Ψ(g, c(g)) = c(g). It is obvious that c(f ) = 0. We prove that c(g) ∈ V. Let x n = R n g (0) ∈ B(r), n ≥ 0. Then x n − c(g) → 0, and it is obvious by induction that x n ∈ X ′ . We have x n+1 = R g (x n ) = J(Ψ(g, x n ) − A(x n ). By (4.2), we get x n+1 ′ ≤ ε 2 (1 + x ′ ), which is equivalent to By induction we easily get x n ′ − ε 2−ε ≤ 0 for all n ≥ 0. Hence x n ′ ≤ ε 2−ε ≤ ε. But the closed ball B ′ (ε) in X ′ is closed in X and x n → c(g) in X. So c(g) ∈ B ′ (ε) ⊂ V.
Let V be a C α neighborhood of id in C α (M, M ) as in Theorem 1.2. Then we may choose a neighborhood V f of id in the Banach manifold C α f (M, M ) of C α and d f -Lipschitz maps on M such that V f ⊂ V, and such that elements in V f are sufficiently d f -Lipschitz close to the identity. Applying Lemma 4.1 to the map Ψ, we get a C 1 neighborhood N of f in M ⊂ Diff 1 (M ) and a function c : N → V f with c(f ) = id such that Ψ(g, c(g)) = gc(g)f −1 = c(g) for every g ∈ N , and c is C 1 as a map N → C 0 (M, M ). It is easy to show that if c(g) is sufficiently d f -Lipschitz close to the identity, then c(g) is a homeomorphism (see [8,9]). So g = c(g)f c(g) −1 . This proves Theorem 1.2.