Stable accessibility is C^{1} dense

Dmitry Dolgopyat and Amie Wilkinson

October 15, 2001

Abstract

We prove that in the space of all C^{r} (r ≥ 1) partially hyperbolic diffeomorphisms, there is a C^{1} open and dense set of accessible diffeomorphisms. This settles the C^{1} case of a conjecture of Pugh and Shub. The same result holds in the space of volume preserving or symplectic partially hyperbolic diffeomorphisms.

Combining this theorem with results in [Br], [Ar] and [PugSh3], we obtain several corollaries. The first states that in the space of volume preserving or symplectic partially hyperbolic diffeomorphisms, topological transitivity holds on an open and dense set. Further, on a symplectic n-manifold (n ≤ 4) the C^{1}-closure of the stably transitive symplectomorphisms is precisely the closure of the partially hyperbolic symplectomorphisms. Finally, stable ergodicity is C^{1} open and dense among the volume preserving, partially hyperbolic diffeomorphisms satisfying the additional technical hypotheses of [PugSh3] .

Introduction

This paper is about the accessibility property of partially hyperbolic diffeomorphisms. We show that accessibility holds for a C^{1} open and dense set in the space of all partially hyperbolic diffeomorphisms, thus settling the C^{1} version of a conjecture of Pugh and Shub [PugSh1]. Partially hyperbolic diffeomorphisms are similar to Anosov diffeomorphisms, in that they possess invariant hyperbolic directions, but unlike Anosov diffeomorphisms, they can also possess invariant directions of non-hyperbolic behavior. Accessibility means that the hyperbolic directions fill up the manifold on a macroscopic scale. Accessibility often provides enough hyperbolicity for a variety of chaotic properties, such as topological transitivity [Br] and ergodicity [PugSh3], to hold. As a consequence, we derive several density results about stable ergodicity and stable transitivity among partially hyperbolic diffeomorphisms.

Let M be a smooth compact, connected and boundaryless Riemannian manifold. A diffeomorphism f : M → M is partially hyperbolic if the tangent bundle to M splits as a T f-invariant sum

T M = E^{u} ⊕ E^{c} ⊕ E^{s},

such that T f uniformly expands all vectors in E^{u} and uniformly contracts all vectors in E^{s}, while vectors in E^{c} are neither contracted as strongly as any vector in E^{s} nor expanded as strongly as any vector in E^{u}. More precisely, for each p ∈ M, there exist 0 < a_{p} < b_{p} < 1 < B_{p} < A_{p} such that:

∥T_{p}f|_{E^{s}}∥ ≤ a_{p} < b_{p} ≤ m(T_{p}f|_{E^{c}}) ≤ ∥T_{p}f|_{E^{c}}∥ ≤ B_{p} < A_{p} ≤ m(T_{p}f|_{E^{u}}),

where m(T) = ∥T^{−1}∥^{−1}. Throughout this paper we assume that both subbundles E^{u} and E^{s} are nontrivial.

A more stringent condition, often called partial hyperbolicity in the literature (cf. [BrPe], [BuPuShWi]) requires that the constants a_{p}, b_{p}, A_{p} and B_{p} be chosen independent of p. Since the results in this paper apply to diffeomorphisms satisfying the weaker condition, to avoid excessive terminology, we will use the term partial hyperbolicity in the broader sense.

A partially hyperbolic diffeomorphism f is accessible if, for every pair of points p, q ∈ M, there is a C^{1} path from p to q whose tangent vector always lies in E^{u} ∪ E^{s} and vanishes at most finitely many times. We say f is stably accessible if every g sufficiently C^{1}-close to f is accessible. We prove here the following theorem.

Main Theorem: For any r ≥ 1, stable accessibility is C^{1} dense among the C^{r}, partially hyperbolic diffeomorphisms of M, volume preserving or not. If M is a symplectic manifold, then stable accessibility is C^{1} dense among C^{r}, symplectic partially hyperbolic diffeomorphisms of M.

Related to the Main Theorem is the result of Niţică and Török [NiTö] that stable accessibility is C^{r}-dense among partially hyperbolic diffeomorphisms with 1-dimensional, integrable center bundle E^{c}. Other results about stable accessibility treat more special classes of diffeomorphisms, such as time-one maps of Anosov flows [BuPuWi], skew products [BuWi1], certain systems where E^{u} ⊕ E^{s} is integrable [ShWi], and systems whose partially hyperbolic splitting is C^{1} [PugSh2].

The Main Theorem has several corollaries. The first corollary concerns the topological transitivity of partially hyperbolic diffeomorphisms and follows immediately from a theorem of Brin [Br]. Denote by PH^{r}(M) the set of C^{r} partially hyperbolic diffeomorphisms of M. If μ and ω are, respectively, Riemannian volume and a symplectic form on M, then set

PH_{μ}^{r}(M) = {f ∈ PH^{r}(M) | f_{∗}(μ) = μ}, and
PH_{ω}^{r}(M) = {f ∈ PH^{r}(M) | f^{∗}(ω) = ω}.

Corollary 0.1 For r ≥ 1, there is a C^{1}-open and dense set of topologically transitive diffeomorphisms in PH_{μ}^{r}(M). If M has a symplectic form ω, then there is a C^{1}-open and dense set of transitive diffeomorphisms in PH_{ω}^{r}(M).

This corollary is false without the volume preservation assumption. Niţică and Török have shown in [NiTö] that there is an open set of accessible nontransitive diffeomorphisms. While it is plausible that for a C^{1} open and dense set of diffeomorphisms in the space PH^{r}(M), there are only finitely many transitivity components, it is not a direct corollary of the Main Theorem.

M.-C. Arnaud has shown in [Ar] that if M is a symplectic 4-manifold, then the stably transitive diffeomorphisms in Diff_{ω}^{r}(M) are partially hyperbolic. (The same result has been announced by J. Xia in arbitrary dimension). Hence there is a complete picture in dimension 4 of the stably transitive diffeomorphisms, which we summarize in the next corollary.

Corollary 0.2 Let M be a symplectic manifold with dim(M) ≤ 4. The C^{1} closure of the stably transitive diffeomorphisms in Diff_{ω}^{r}(M) coincides with the C^{1} closure of the partially hyperbolic ones.

In other words, invariant tori are essentially the only obstacle for topological transitivity in the symplectic category, at least if dim(M) ≤ 4. We conjecture that the same is true in the volume preserving case.

Conjecture 0.3 In the space of volume preserving diffeomorphisms, the C^{1}- closure of the stably transitive diffeomorphims coincides with the closure of the diffeomorphisms admitting a dominated splitting.

For a discussion of the dominated splitting condition and some results related to Conjecture 0.3 see [Vi]. Even though the results of this paper could be useful in attacking this conjecture some other ideas (possibly ones from the paper [BonDi]) are necessary to solve this problem. Here we note only that in [BV] a volume preserving example is presented which is stably transitive yet not partially hyperbolic. A. Tahzhibi has announced a proof that these example are in fact stably ergodic.

Another corollary of the Main Theorem concerns ergodicity of f ∈ PH_{μ}^{r}(M). Pugh and Shub proved the following theorem:

Theorem 0.4 [PugSh3, Theorem A] Let f ∈ PH_{μ}^{2}(M). If f is center bunched, dynamically coherent, and essentially accessible, then f is ergodic.

Thus we also have the corollary:

Corollary 0.5 Among the center bunched, stably dynamically coherent diffeomorphisms in PH_{μ}^{2}(M), stable ergodicity is C^{1} open and dense.

Theorem 0.4 refers to partially hyperbolic diffeomorphisms in the stronger sense described earlier, but recently Burns and Wilkinson [BuWi2] have shown that these results extend to the larger class of partially hyperbolic diffeomorphisms described in this paper (satisfying additional center bunching conditions). For a description of examples of diffeomorphisms satisfying the conditions “center bunched” and “stably dynamically coherent” see the survey paper [BuPuShWi]. In particular, the corollary implies that there is a C^{1}-open neighborhood U ⊂ PH_{μ}^{2}(M) of f in which stable ergodicity is C^{1}-open and dense, where f is the time-t map of an Anosov flow, a compact group extension of an Anosov diffeomorphism, an ergodic automorphism of a torus or nilmanifold, or a partially hyperbolic translation on a compact homogeneous space.

This paper arose out of an attempt to prove the following conjecture of Pugh and Shub.

Conjecture 0.6 [PugSh2, Conjecture 4] and [PugSh3, Conjecture 2] Stable accessibility is C^{r} - dense in both PH^{r}(M) and PH_{μ}^{r}(M).

In the spirit of Theorem 0.4, Pugh and Shub also conjectured:

Conjecture 0.7 [PugSh3, Conjecture 3] A partially hyperbolic C^{2} volume preserving diffeomorphism with the essential accessibility property is ergodic.

Finally, combining Conjectures 0.6 and 0.7, they conjectured:

Conjecture 0.8 [PugSh3, Conjecture 4] Stable ergodicity is C^{r} - dense in PH_{μ}^{r}(M).

As with Theorem 0.4, these conjectures refer to the narrower class of partially hyperbolic diffeomorphisms described above, but in light of the results in [BuWi2] and this paper, it seems reasonable to extend them to the class under consideration here.

The question of accessibility is closely related to problems in control theory (see, e.g. [Lo]). In fact, analogous density theorems in control theory initially suggested the Conjectures 0.6 and 0.7. The sole reason that the results in control theory cannot be directly transported to this setting is that we do not perturb the bundles E^{u} and E^{s} directly, but rather the diffeomorphism f. We’d like to be able to say that a specific perturbation of f has a specific effect on E^{u} and E^{s}. What makes this difficult is that E^{s}(p) and E^{u}(p) are determined by the entire forward and backward orbit of p, respectively; a perturbation will have various effects along the length of this orbit, some desirable and others not.

The key observation that permits a measure of control is that the effects of the perturbation are greatest along the first few iterates of p. To maximize our control over the bundles E^{u} and E^{s}, we isolate regions where we need local accessibility and localize the perturbation to these regions. Choosing the support of the perturbation to be highly non-recurrent then minimizes undesirable “noisy” effects of the perturbations. The trade-off is that the desirable effects of the pertubations are necessarily quite small. Nonetheless, with the right C^{1}-small perturbation, the desirable effects outweigh the undesirable ones and we obtain accessibility. Similar perturbations are found in [PP].

It appears that a localized C^{2}-small perturbation cannot achieve this, and so the techniques in this paper do not extend to the C^{2} setting. New techniques would be required to prove Conjectures 0.6 and 0.7.

Here is how the proof of the Main Theorem goes. Let f ∈ PH^{r}(M) have partially hyperbolic splitting T M = E^{u} ⊕ E^{c} ⊕ E^{s}. By [BrPe, HiPuSh] E^{u} and E^{s} are tangent to the leaves of continuous foliations which are denoted W^{u} and W^{s} and are called the unstable and stable foliations respectively. A us-path for f is a path γ : [0, 1] → M consisting of a finite number of consecutive arcs — called legs — each of which is a curve that lies in a single leaf of W^{u} or W^{s}. It is easy to see that f is accessible if and only if for all p, q ∈ M, there is a us-path for f from p to q.

To prove the Main Theorem, we first find a collection of disjoint disks in M. Each disk is approximately tangent to the center direction E^{c}. We choose this collection large enough so that f is accessible, modulo these disks. More precisely, for every p, q ∈ M, there is a finite sequence of us-paths for f, the first path originating at p and ending in one of the disks, the last path originating in a disk and ending at q. The intermediate paths all begin and end in disks, each path beginning in the disk where the previous path ends. We then perturb f in a small neighborhood of these disks. We can arrange that if this neighborhood and the C^{0}-size of the pertubation are both sufficiently small, then the perturbed system will still be accessible, modulo the same collection of disks. It is not hard to see that any additional C^{1}-small perturbation will preserve this property.

Under the C^{r}-dense assumption that the fixed points of f^{k} are isolated, for all k ≥ 1, we can choose these disks to be very small and their union highly non-recurrent. This is Lemma 1.2. We show in Lemma 1.1 that it is then possible to perturb f in a neighborhood of these disks by a C^{1}-small perturbation to obtain a stably accessible g. We prove stable accessibility by showing that any two points in a given disk can be connected by a us-path for g, and for any small perturbation of g. Since any small perturbation of g is already accessible modulo these disks, this gives stable accessibility.

Lemma 1.1 is the only place where it is essential that the perturbation be only C^{1}-small. When we examine the effect of perturbing f on E^{u} and E^{s}, we find that in C^{1}, the contribution to E^{u}(p) and E^{s}(p) of the perturbation near p is larger than the combined contributions along the rest of the orbit of p; this is not true in C^{2}. Therefore, a more complicated analysis, taking into account the first several returns, is needed to establish the analogue of our result in the C^{2}-setting.

1 Proof of the Main Theorem

Proof: We first prove the Main Theorem in the case where f preserves a smooth volume μ. The proof is easily modified to the non volume preserving case. In the final section we describe how to modify the proof for the symplectic case.

Let f ∈ PH_{μ}^{r}(M) and δ > 0 be given. Our goal is to find a stably accessible g ∈ Diff_{μ}^{r}(M) with d_{C^{1}}(f, g) < δ. We give some preliminary definitions and notation.

Let P(M) be the collection of all subsets of M. We say that f is accessible on X ∈ P (M) if, for every p, q ∈ X, there is a us-path for f from p to q. The diffeomorphism f is accessible modulo X ⊂ P(M) if, for every p, q ∈ M, there exist X_{1}, …, X_{r} ∈ X and us-paths for f: from p to X_{1}, from q to X_{r}, and from X_{m} to X_{m+1}, for m = 1, …, r−1. We say that f is uniformly accessible modulo X if f is accessible modulo X and, further, there is a number N such that the us-paths in the previous definition can be chosen to have length less than N and to have fewer than N legs.

A property of a diffeomorphism f is said to hold stably if it also holds for all g ∈ Diff^{1}(M) sufficiently C^{1}-close to f. Clearly if f is stably accessible modulo {X_{1}, …, X_{k}} and f is stably accessible on each X_{i}, then f is stably accessible.

Define a function R : P(M) → N ∪ {∞} as follows. For X ∈ P(M), let R(X) be the smallest J ∈ N ∪ {∞} satisfying:

f^{i}(X) ∩ X ≠ ∅, with |i| = J + 1.  (1)

Note that R(B_{ρ}(p)) → per(p), as ρ → 0, where we set per(p) = ∞ if p is not periodic.

We next fix a system of local charts on M. We will on several occasions refer to the orthogonal splitting R^{n} = T_{v}R^{n} = R^{u} ⊕ R^{c} ⊕ R^{s}, where a = dim(E^{a}).

Let B^{n}(v, ρ) denote the ball of radius ρ about v ∈ R^{n} with respect to the sup norm on coordinates. More generally, we will use the notation B^{a}(v, ρ), where a = u, c, s, c + u, c + s, or u + s, to denote the sup-norm ball of radius ρ about v in the affine space v + R^{a}.

Applying Moser’s theorem on the equivalence of volume forms [Mo] we obtain, for any p ∈ M, a C^{∞} map

φ_{p} : B^{n}(0, 1) → M

such that

1. φ_{p}(0) = p,

2. T_{0}φ_{p} sends the splitting T_{0}R^{n} = R^{u} ⊕ R^{c} ⊕ R^{s} to the splitting T_{p}M = E^{u} ⊕ E^{c} ⊕ E^{s}, 3. φ_{p} sends divergence-free vector fields to divergence-free vector fields.

4. p ↦ φ_{p} is a uniformly continuous map from M to C^{1}(B^{n}(0, 1), M). The dependence of φ_{p} on f is also continuous.

([Mo] gives maps satisfying 1, 3 and 4. 2 can be achived by precomposing with a linear map.)

Since we do not assume that E^{c} is tangent to a foliation, we will work with approximate center manifolds. For ρ < 1 and p ∈ M, let

V_{ρ}(p) = φ_{p}(B^{c}(0, ρ)).

We refer to V_{ρ}(p) as a c-admissible disk with center p and radius ρ and write r(V_{ρ}(p)) = ρ. If D is a c-admissible disk with center p and radius ρ, then for β ∈ (0, 1), we denote by βD the c-admissible disk with center p and radius βρ.

A c-admissible family is a finite collection of pairwise disjoint, c-admissible disks. If D is a c-admissible family, and β < 1, then let

βD = {βD | D ∈ D},
|D| = ⋃_{D∈D} D,
r(D) = sup_{D∈D} r(D), and
R(D) = R(|D|).

We have the following lemma.

Lemma 1.1 (Accessibility on central disks) Let f ∈ PH_{μ}^{r}(M) and δ > 0 be given. Then there exists J > 0 with the following property.

If D is a c-admissible family with r(D) < J^{−1} and R(D) > J, then for all σ > 0 and β ∈ (0, 1), there exists g ∈ Diff_{μ}^{r}(M) such that:

1. d_{C^{1}}(f, g) < δ,

2. d_{C^{0}}(f, g) < σ,

3. For each D ∈ D, g is stably accessible on βD.

We may assume that the fixed points of f^{k} are isolated, for all k ≥ 1; this property is C^{r}-dense in Diff_{μ}^{r}(M). Under this additional assumption we have the following lemma.

Lemma 1.2 (Accessibility modulo central disks) Let f ∈ PH_{μ}^{r}(M) be given. Assume that the fixed points of f^{k} are isolated, for all k ≥ 1. Then for every J > 0 there exists a c-admissible family D such that:

1. r(D) < J^{−1},

2. R(D) > J,

3. f is uniformly accessible modulo \frac{1}{2}D.

Lemma 1.3 (Persistence of accessibility modulo D) There exists δ_{0} > 0 so that, given δ < δ_{0}, a c-admissible family D with f uniformly accessible modulo \frac{1}{2}D, and β ∈ (\frac{1}{2}, 1), the following holds.

There exists σ > 0 such that any g satisfying

1. d_{C^{1}}(f, g) < δ,

2. d_{C^{0}}(f, g) < σ,

is accessible modulo βD (and hence g is stably accessible modulo βD, since 1. and 2. are open conditions).

The proof of the Main Theorem now follows from Lemmas 1.1, 1.2 and 1.3. Let f and δ be given. After a C^{r}-small perturbation, we may assume that the fixed points of f^{k} are isolated, for all k. We may assume that δ < δ_{0}, where δ_{0} is given by Lemma 1.3

Choose J according to Lemma 1.1. By Lemma 1.2, there exists a cadmissible family D, with R(D) > J and r(D) < J^{−1}, such that f is accessible modulo \frac{1}{2}D.

Now fix β ∈ (\frac{1}{2}, 1), and choose σ according to Lemma 1.3. Applying Lemma 1.1 we obtain a diffeomorphism g ∈ Diff_{μ}^{r}(M), with d_{C^{1}}(f, g) < δ and d_{C^{0}}(f, g) < σ, such that g is stably accessible on βD, for each D ∈ D. By Lemma 1.3, g is also stably accessible modulo βD. Thus, g is stably accessible.♢

The proofs of Lemmas 1.2 and 1.3 are given in the next section, and the proof of Lemma 1.1 is given in Section 3.

The arguments of Section 2 become simpler if E^{c}(f) is integrable. In that case, one can work with central disks instead of c-admissible ones. To construct a family of central disks satsifying the conditions of Lemma 1.2 one should take a small ε > 0, choose an ε–net {p_{j}} and let D be the union of unit central disks centered at p_{j}. If there are some i, j < Card(D) and n < J such that f^{n}D(p_{i}) ⋂ D(p_{j}) ≠ ∅ then one can remove this intersection by arbitrary small shift of p_{i} in an unstable direction (this is possible even if i = j since the periodic leaves of f are isolated by partial hyperbolicity). In case E^{c} is not integrable the proof becomes significantly more complicated since one has to work with disks which are almost tangent to E^{c}, but the idea remains the same. As it is mentioned in the introduction the most difficult part of the proof is Section 3 where the abundance of C^{1}-perturbations is crucial. Therefore many readers would find it helpful to skip Section 2 during the first reading returning to it after mastering Section 3.

2 Global Accessibility

In this section we prove Lemmas 1.2 and 1.3.

Proof of Lemma 1.2: Let J be given. Let A = {p ∈ M | per(p) ≥ J +2}. Since the fixed points of f^{J+2} are isolated, there exist x_{1}, …, x_{m} ∈ M such that A = M \ {x_{1}, …, x_{m}}.

For ρ > 0, let U_{ρ}(p) be the image of the ball B^{n}(0, ρ) under φ_{p}. The proof of the following lemma is straightforward.

Lemma 2.1 If r > 0 is sufficiently small, then every p ∈ ⋃ U_{r}(x_{i}) can be connected to a point in M \ ⋃ U_{r}(x_{i}) by a us-path with one leg.

Choose r according to this lemma, and let A_{r} = M \ ⋃ U_{r}(x_{i}). Assume r is small enough that A_{r} is connected. Since A_{r} is compact and contained in A, there exists ρ_{0} < 1/(2J) such that R(U_{2ρ_{0}}(q)) > J, for every q ∈ A_{r}.

The next lemma follows from the uniform continuity of the invariant splitting for f, and we omit its proof.

Lemma 2.2 There exists K > 1 so that, for ρ_{0} sufficiently small, for all p ∈ M, and for every q_{1}, q_{2} ∈ U_{ρ_{0}/K}(p), there is a us-path with ≤ 2 legs from q_{1} to some point in V_{ρ_{0}}(q_{2}).

The next lemma is key.

Lemma 2.3 Let K > 1. If ρ_{0} is sufficiently small, then there exist a cover of A_{r} by finitely many neighborhoods U_{1}, …, U_{k} of the form U_{i} = U_{ρ_{0}/K}(q_{i}), and, for i = 1, …, k, points p_{i} ∈ U_{i} such that

V_{2ρ_{0}}(p_{i}) ∩ V_{2ρ_{0}}(p_{j}) = ∅,

for i ≠ j, and

V_{2ρ_{0}}(p_{i}) ∩ f^{m} (V_{2ρ_{0}}(p_{j})) = ∅,

for all i, j and 0 < |m| ≤ J.

Before proving Lemma 2.3, we finish the proof of Lemma 1.2.

Let U_{1}, …, U_{k} and p_{1}, …, p_{k} be given by Lemma 2.3. For i = 1, …, c, let D_{i} = V_{2ρ_{0}}(p_{i}). Note that the collection D = {D_{1}, …, D_{k}} satisfies conclusions 1. and 2. of Lemma 1.2, if ρ_{0} is sufficiently small.

Lemma 2.2 implies that, for every p ∈ B_{i}, there is a us-path with ≤ 2 legs from p to some point in \frac{1}{2}D_{i}. It follows that, whenever B_{i} ∩ B_{j} ≠ ∅, there is a us-path with ≤ 4 legs from some point on \frac{1}{2}D_{i} to some point in \frac{1}{2}D_{j}. Since A_{r} is connected and the balls B_{i}, …, B_{k} cover A_{r}, we obtain, for any i, j, a sequence of disks D_{a_{0}} = D_{i}, D_{a_{1}}, …, D_{a_{l}} = D_{j}, such that \frac{1}{2}D_{a_{m}} is connected to \frac{1}{2}D_{a_{m+1}} by a us-path, for m = 0, …, l − 1. Then for any p, q ∈ A_{r}, there are a sequence of disks D_{b_{0}}, …, D_{b_{s}} and us-paths: from p to \frac{1}{2}D_{b_{0}}, from q to \frac{1}{2}D_{b_{s}}, and from \frac{1}{2}D_{b_{r}} to \frac{1}{2}D_{b_{m+1}}, for m = 0, …, s − 1. The length and number of legs of these paths is clearly bounded. Since any point in M \A_{r} = ⋃ B_{r}(x_{i}) can be connected to a point in A_{r} by a us-path with one leg, it follows that f is uniformly accessible modulo \{\frac{1}{2}D_{1}, …, \frac{1}{2}D_{k}\}. This proves 3., completing the proof of Lemma 1.2.♢

Proof of Lemma 2.3: We start with a simple Besicovich-type covering lemma.

Lemma 2.4 (Covering lemma) For any C > 0 there exists an integer N > 0 such that, for any compact set A ⊆ M, and for ρ > 0 sufficiently small, there exist q_{1}, …, q_{k} ∈ A with the following properties, for i = 1, …, k:

• A ⊆ B_{ρ}(q_{1}) ∪ ⋯ ∪ B_{ρ}(q_{k}), and

• #{j | B_{Cρ}(q_{i}) ∩ B_{Cρ}(q_{j}) ≠ ∅} ≤ N.

Proof of Lemma 2.4: On the manifold M, there exists a constant K > 0 such that for every ρ < 1 and every p ∈ M, the volume of the ball B_{ρ}(p) lies between ρ^{n}/K and Kρ^{n}. Let N = (4C + 2)^{n}K^{2}; this is an upper bound on the number of disjoint balls of radius ρ/2 that can fit inside a ball of radius (2C + 1)ρ.

Let A and ρ be given. Let S_{ρ} ⊂ M be a maximal ρ-separated subset of A. Such a set exists by compactness of A. We claim that S_{ρ} is also ρ-spanning.

If not, then there exists x ∈ A such that d(x, y) > ρ, for all y ∈ A. But this contradicts maximality of S_{ρ}. Hence, if q_{1}, …, q_{k} are the elements of S_{ρ}, then

A ⊆ B_{ρ}(q_{1}) ∪ ⋯ ∪ B_{ρ}(q_{k}).

For p ∈ S_{ρ}, let N(p) be the set of q ∈ S_{ρ} such that

B_{Cρ}(p) ∩ B_{Cρ}(q) ≠ ∅.

For each q ∈ N(p), the distance from p to q is less than 2Cρ, and so the ball B_{ρ/2}(q) is contained in B_{(2C+1)ρ}(p). Since S_{ρ} is ρ-separated, the balls B_{ρ/2}(q) and B_{ρ/2}(q′) are disjoint, for distinct q, q′ ∈ N(p). The cardinality of N(p) is thus bounded by N, which completes the proof.♢

The sets U_{ρ}(p) are uniformly comparable to round balls B_{ρ}(p), and the maps {f^{m}, |m| ≤ J} distort distances by a bounded factor. Thus Lemma 2.4 implies the following.

Corollary 2.5 (Strengthened covering lemma) Let C, J > 0 be given. There exists an integer N > 0 such that, for any compact set A ⊆ M, and for any ρ > 0, there exist q_{1}, …, q_{k} ∈ A with the following properties, for i = 1, …, k:

• A ⊆ U_{ρ}(q_{1}) ∪ ⋯ ∪ U_{ρ}(q_{k}), and

• #{j | U_{Cρ}(q_{i}) ∩ f^{m} (U_{Cρ}(q_{j})) ≠ ∅, for some |m| < J} ≤ N.

We now return to the proof of Lemma 2.3. To simplify notation, let ρ_{1} = ρ_{0}/K and let ρ_{2} = 4ρ_{0}, where ρ_{0} is defined after the statement of Lemma 2.1. Thus ρ_{1} < ρ_{0} < ρ_{2} In this notation, we have that R(U_{ρ_{2}}(p)) > J, for all p in A_{r}.

For p ∈ M and ρ > 0, let T_{ρ}(p) be the connected component of p in φ_{p}(B^{u+s}(0, ρ)). For d(p, q) small enough, the maps φ_{q}^{−1}φ_{p} distort the Euclidean structure by a factor ≤ 1.5. Assume that ρ_{2} is small enough that this distortion bound holds whenever p, q ∈ U_{ρ_{2}}(z), for any z. From this we obtain that for all p ∈ M and all q ∈ T_{ρ_{1}}(p),

V_{2ρ_{0}}(q) ⊂ U_{ρ_{1}+3ρ_{0}}(p) ⊂ U_{ρ_{2}}(p).  (2)

We now apply Corollary 2.5 with C = 4K, A = A_{r}, and ρ = ρ_{1}. By Corollary 2.5, there exists N > 0 and q_{1}, …, q_{k} ∈ A_{r} such that • A_{r} = U_{ρ_{1}}(q_{1}) ∪ ⋯ ∪ U_{ρ_{1}}(q_{k}), and

• #{j | U_{ρ_{2}}(q_{i}) ∩ f^{m} (U_{ρ_{2}}(q_{j})) ≠ ∅, for some |m| < J} ≤ N.

For i = 1, … k, let U_{i} = U_{ρ_{1}}(q_{i}). The neighborhoods U_{1}, …, U_{k} cover A_{r}.

We choose p_{1}, … p_{k} inductively. Let p_{1} = q_{1}. Since V_{2ρ_{0}}(p_{1}) ⊂ U_{ρ_{2}}(p_{1}), and R(U_{ρ_{2}}(p_{1})) > J, we have that

V_{2ρ_{0}}(p_{1}) ∩ f^{m}(V_{2ρ_{0}}(p_{1})) = ∅,

for 0 < |m| ≤ J.

Fix i > 1, and suppose that the points p_{1}, …, p_{i−1} have already been chosen. We want to choose p_{i} so that

V_{2ρ_{0}}(p_{i}) ∩ f^{m}(V_{2ρ_{0}}(p_{i})) = ∅,

for 0 < |m| ≤ J, and

V_{2ρ_{0}}(p_{i}) ∩ f^{m}(V_{2ρ_{0}}(p_{j})) = ∅,

for 0 ≤ |m| ≤ J and j < i. The first of these two properties is satisfied if we choose p_{i} so that V_{2ρ_{0}}(p_{i}) ⊆ U_{ρ_{2}}(q_{i}). By (2), this is turn will hold if we choose p_{i} ∈ T_{ρ_{1}}(q_{i}).

Hence, we would like to find p_{i} ∈ T_{ρ_{1}}(q_{i}) such that

V_{2ρ_{0}}(p_{i}) ∩ f^{m}(V_{2ρ_{0}}(p_{j})) = ∅,  (3)

for 0 ≤ |m| ≤ J and j < i. The neighborhood U_{ρ_{2}}(q_{i}) meets at most N sets of the form f^{m} (U_{ρ_{2}}(q_{j})), for m between −J and J. Thus, #J_{i} ≤ N, where J_{i} is the collection of all (j, m) with j < i, |m| < J, and

U_{ρ_{2}}(q_{i}) ∩ f^{m}(U_{ρ_{2}}(q_{j})) ≠ ∅.

For q ∈ M and |m| ≤ J, let V_{ρ}^{m}(q) be the connected component of f^{m}(q) in U_{ρ}(f^{m}(q)) ∩ f^{m}(V_{1}(q)). There exists C_{0} ≥ 1 such that, for all p, q ∈ M,

• V_{C_{0}ρ_{2}}^{m}(q) ⊇ f^{m}(V_{2ρ_{0}}(q)), and

• if U_{ρ_{2}}(p) ∩ f^{m}(U_{ρ_{2}}(q)) ≠ ∅, and ρ_{0} is sufficiently small, then V_{C_{0}ρ_{2}}^{m}(q) intersects T_{C_{0}ρ_{1}}(p) in exactly one point.

(It is not hard to see that C_{0} can be chosen to depend only on J, on the Riemannian structure, and on ∥T f|_{E^{c}}∥.) For (j, m) ∈ J_{i}, let p′_{j,m} be the point of intersection of V_{C_{0}ρ_{2}}^{m}(p_{j}) and T_{C_{0}ρ_{1}}(q_{i}):

{p′_{j,m}} = V_{C_{0}ρ_{2}}^{m}(p_{j}) ∩ T_{C_{0}ρ_{1}}(q_{i}).

Consider the collection of these points

P_{i} = {p′_{j,m} | (j, m) ∈ J_{i}} ⊂ T_{C_{0}ρ_{1}}(q_{i}).

The points in φ_{q_{i}}^{−1}(P_{i}) lie on B^{u+s}(0, C_{0}ρ_{1}) ⊂ R^{n}. By elementary Euclidean geometry, there exists C_{1} > 0 such that, for any ρ > 0, and any finite collection of points Q ⊂ B^{u+s}(0, C_{0}ρ), there is a point v ∈ B^{u+s}(0, C_{0}ρ) whose distance to the points in Q is at least ρ/(C_{1}#Q).

Applying this fact to the points in φ_{q_{1}}^{−1}(P_{i}), we find that there exists a point p_{i} ∈ T_{ρ_{1}}(q_{i}) ⊂ U_{i}, such that, for all p ∈ P_{i},

∥φ_{q_{i}}^{−1}(p_{i}) − φ_{q_{i}}^{−1}(p)∥ ≥ ρ_{1}/(C_{1}#P_{i}) ≥ ρ_{1}/(C_{1}N).  (4)

We claim that if ρ_{0} is sufficiently small, then p_{i} satisfies (3); that is, for all j < i and |m| ≤ J,

V_{2ρ_{0}}(p_{i}) ∩ f^{m} (V_{2ρ_{0}}(p_{j})) = ∅.

Clearly the claim is true for those (m, j) such that U_{ρ_{2}}(q_{i})∩f^{m} (U_{ρ_{2}}(q_{j})) = ∅, so suppose that (m, j) ∈ J_{i}. We show that

V_{2ρ_{0}}(p_{i}) ∩ V_{C_{0}ρ_{2}}^{m}(p_{j}) = ∅,

which clearly implies the result.

We shall view everything in R^{n}. Under the map φ_{p_{i}}^{−1}, the sets V_{2ρ_{0}}(p_{i}) and V_{C_{0}ρ_{2}}^{m}(p_{j}) lift, respectively, to B^{c}(0, 2ρ_{0}) and a set we’ll call W. We show that B^{c}(0, 2ρ_{0}) and W are disjoint, for ρ_{0} sufficiently small.

The set V_{C_{0}ρ_{2}}^{m}(p_{j}) is a C^{1} disk, tangent at f^{m}(p_{j}) to E^{c}(f^{m}(p_{j})). Thus W is a C^{1} disk, tangent at a point w_{1} ∈ W to the uniformly continuous distribution T φ_{p_{i}}^{−1}(E^{c}). Furthermore, the distribution T φ_{p_{i}}^{−1}(E^{c}) coincides at p_{i} with R^{c}.

By the distortion bounds, the diameter of W is on the order of ρ_{2}:

diam (W) ≤ 3C_{0}ρ_{2}.  (5)

Let w_{2} = φ_{p_{i}}^{−1}(p′_{j,m}) ∈ W. Combining the distortion bounds with (4), we obtain that

∥w_{2}∥ = ∥φ_{p_{i}}^{−1}(p_{i}) − φ_{p_{i}}^{−1}(p′_{j,m})∥ ≥ 2ρ_{1}/(3C_{1}N).  (6)

All of these statements – about the C^{1}-smoothness of W, the continuity of the distribution T φ_{p_{i}}^{−1}E^{c}, etc. – hold uniformly over p_{i}, ρ_{0} and |m| < J. Thus, to summarize the preceding remarks, we have a constant C_{2} > 0, and functions θ_{1}, θ_{2} : R_{+} → R_{+}, all independent of p_{i}, ρ_{0} and m, such that W is contained in the graph of a C^{1} function F : B^{c}(0, C_{2}ρ_{0}) → R^{u+s}, with:

1. ∥DF (x_{1})∥ ≤ θ_{1}(∥x_{1}∥), for some x_{1} ∈ B^{c}(0, C_{2}ρ_{0}), (x_{1} corresponds to the point w_{1} ∈ W),

2. ∥F (x_{2})∥ ≥ ρ_{0}/C_{2}, for some x_{2} ∈ B^{c}(0, C_{2}ρ_{0}), (x_{2} corresponds to the point w_{2}),

3. ∥F (y)−F (x)−DF (x)(y −x)∥ ≤ θ_{2}(∥y −x∥), for all x, y ∈ B^{c}(0, C_{2}ρ_{0}),

4. lim_{r→0} θ_{1}(r) = 0, and lim_{r→0} θ_{2}(r)/r = 0.

We claim that if ρ_{0} is small enough, then ∥F (x)∥ > 0, for all x ∈ B^{c}(0, C_{2}ρ_{0}). This implies that W is disjoint from B^{c}(0, 2ρ_{0}), which is the desired result. By 3., we have that for all x ∈ B^{c}(0, 2ρ_{0}),

∥F (x) − F (x_{1})∥ ≤ ∥DF (x_{1})(x − x_{1})∥ + θ_{2}(∥x − x_{1}∥)
≤ 2C_{2} ρ_{0}θ_{1}(2C_{2}ρ_{0}) + θ_{2}(2C_{2}ρ_{0}).

By the triangle inequality,

∥F (x)∥ ≥ ∥F (x_{2})∥ − ∥F (x_{2}) − F (x_{1})∥ − ∥F (x_{1}) − F (x)∥
≥ ρ_{0}/C_{2} − 4C_{2}ρ_{0} θ_{1}(2C_{2}ρ_{0}) − 2θ_{2}(2C_{2}ρ_{0}).

If ρ_{0} is sufficiently small, then this quantity is positive.♢

Proof of Lemma 1.3: We want to pass from infinitesimal conditions given in the definition of partial hyperbolicity to local conditions. To this end, let

\overline{a}_{p}(r) = max_{q∈B_{r}(p)} ∥T_{q}f|_{E^{s}(q)}∥,

\overline{b}_{p}(r) = min_{q∈B_{r}(p)} m(T_{q}f|_{E^{c}(q)}),

\overline{B}_{p}(r) = max_{q∈B_{r}(p)} ∥T_{q}f|_{E^{c}(q)}∥,

\overline{A}_{p}(r) = min_{q∈B_{r}(p)} m(T_{q}f|_{E^{u}(q)}).

By continuity of T f we can choose r_{0} > 0 and θ < 1 so that \overline{a}_{p}(r_{0}) < θ\overline{b}_{p}(r_{0}) and \overline{A}_{p}(r_{0}) < θ\overline{B}_{p}(r_{0}). From now on we will fix this r_{0} and write \overline{a}_{p} etc. instead of \overline{a}_{p}(r_{0}) etc.

Let F_{1} and F_{2} be two continuous foliations with C^{1} leaves on M. We say that F_{2} is ε (C^{0}–) close to F_{1} if given any p, q on the same leaf of F_{1} with the leafwise distance at most 1 apart, the F_{2} leaf passing through p intersects the ε ball centered at q.

It is clear that there exists ε > 0 such that g is stably accessible modulo βD provided that W_{g}^{u} is ε close to W_{f}^{u} and W_{g}^{s} is ε close to W_{f}^{s}. So we need to show that given ε > 0 there is σ > 0 such that the conditions of Lemma 1.3 with this σ imply ε closeness of dynamical foliations of g to those of f. Namely we prove that for all p, q ∈ B_{r_{0}/2}^{s}(f, p), the intersection W^{s}(g, p) ⋂ B_{ε}(q) is non-empty. (Thus, 1 in the initial definition is replaced by r_{0}/2 but this is sufficient because the unit ball can be covered by a finite number of r_{0}/2-balls.) Let α_{p}(n) denote

α_{p}(n) = α_{p}α_{fp} … α_{f^{n−1}p}

where α is either of \overline{a}, \overline{A}, \overline{b}, \overline{B}.

Partial hyperbolicity implies that given η ∈ (θ, 1) there is a continuous cone family K^{cu} around E^{u} ⊕ E^{c} such that

(a) T f(K^{cu}(p)) ⊂ K^{cu}(fp),

(b) K^{cu} is uniformly transverse to E_{f}^{s}, and

(c) for any v ∈ K^{cu}(p)

∥T f(v)∥ > \overline{a}_{p}η∥v∥.

For δ_{0} sufficiently small, K^{cu} will also satisfy (a)–(c) if f is replaced by any g such that d_{C^{1}}(f, g) ≤ δ_{0}. Let q ∈ W_{f}^{s}(p) and let d_{W^{s}}(p, q) ≤ r_{0}/2. Then d(f^{N}p, f^{N}q) ≤ \overline{a}_{p}(N). Let V be a topological disk of dimension dim(E^{u}⊕E^{c}) passing through q and such that T V belongs to K^{cu} (for example, we could take V = φ_{q}(B^{u+c}(0, 1))).

Given n we can find σ so small that d_{C^{0}}(f, g) < σ implies that

d(g^{n}p, g^{n}V) < 2\overline{a}_{p}(n).

Since g^{n}V is uniformly transverse to E^{s} there exists C = C(f) such that W_{g}^{s}(g^{n}p) ⋂ g^{n}V contains a point z with d(g^{n}q, z) ≤ C\overline{a}_{p}(n). Hence g^{−n}z ∈ W^{s}(p) and d(q, g^{n}z) ≤ Cη^{n}. Thus, if n is large enough, W_{g}^{s} is ε−close to W_{f}^{s}. ♢

3 Local accessibility

Proof of Lemma 1.1: Let f and δ < δ_{0} be given. Assume δ_{0} is small enough that any g within δ_{0} of f in the C^{1}-metric remains partially hyperbolic, with constants a, b. Since our perturbations are local, it is convenient to adapt the structures we use to a neighborhood of a point p. To each p ∈ M we shall associate:

1. a neighborhood U_{p} = φ_{p}(B^{n}(0, 1)),

2. a C^{∞} Riemann structure g_{p} on U_{p} with path metric d_{p}, isometric under φ_{p}^{−1} to the Euclidean metric on B^{c}(0, 1),

3. a C^{∞} splitting T U_{p} = Ẽ_{p}^{u} ⊕ Ẽ_{p}^{c} ⊕ Ẽ_{p}^{s} = T φ_{p}(R^{u} ⊕ R^{c} ⊕ R^{s}), that agrees with E^{u} ⊕ E^{c} ⊕ E^{s} at p,

4. C^{∞} foliations W̃_{p}^{u}, W̃_{p}^{s}, W̃_{p}^{cu}, W̃_{p}^{cs} of U_{p}, tangent to the corresponding subbundles of the C^{∞} splitting in 3.,

5. for i = 1, …, c, partial flows ζ_{t}^{i} : U_{p} → U_{p} tangent to the leaves of W̃^{c},

6. partial flows τ_{t}^{u} : U_{p} → U_{p} and τ_{t}^{s} : U_{p} → U_{p} tangent to the leaves of W̃^{u}, W̃^{s}, respectively.

We describe the construction of 5. and 6. in more detail. Let {e_{1}, …, e_{c}} be an orthonormal basis for the R^{c} factor in the splitting R^{n} = R^{u}⊕R^{c}⊕R^{s}. For i = 1, … c, define the partial flows ζ_{t}^{i} : B → B by

ζ_{t}^{i}(φ_{p}(v)) = φ_{p}(v + te_{i}).

Similarly, fix unit vectors w^{u} and w^{s} tangent to the R^{u} and R^{s} factors in the splitting R^{n} = R^{u} ⊕R^{c} ⊕R^{s}, and define the partial flows τ_{t}^{u}, τ_{t}^{s} : B → B, by

τ_{t}^{u}(φ_{p}(v)) = φ_{p}(v + tw^{u}), and
τ_{t}^{s}(φ_{p}(v)) = φ_{p}(v + tw^{s}).

Note that τ_{t}^{u} (resp. τ_{t}^{s}) sends W̃^{cs} leaves (resp. W̃^{cu} leaves) to other W̃^{cs} leaves, and between W̃^{cs} leaves is the exactly the W̃^{u} (resp. W̃^{s}) holonomy map. Note that, where defined, τ_{−t}^{s}τ_{−t}^{u}τ_{t}^{s}τ_{t}^{u} is the identity. This expresses the fact that W̃^{u} and W̃^{s} are jointly integrable. In Subsection 3.2, we will use the partial flows τ^{u}, τ^{s}, ζ_{t}^{1}, …, ζ_{t}^{c} to define g.

The next lemma follows directly from the uniform continuity of φ_{p}.

Lemma 3.1 The structures described in 1.-6. are uniform over p ∈ M and over g sufficiently C^{1}-close to f. For all p ∈ M, the structure g_{p} is uniformly comparable to the original Riemann structure on U_{p}.

Since all estimates involving the Riemann structure on M in this paper are local, uniform over p ∈ M, any statement about the Riemann structure becomes valid for g_{p} by introducing a multiplicative constant. We will therefore be deliberately ambiguous in our notation, using d interchangeably for the Riemannian metric and the local metric d_{p}. Also, when the point is clear from the context, we will drop the subscript p in describing the various structures.

3.1 A criterion for stable accessibility

We describe here a condition on the foliations W_{g}^{u}, W_{g}^{s} that implies that g is stably accessible on a c-admissible disk for f.

Let D be a c-admissible disk for f centered at p ∈ M, and let ρ = r(D). Let m = m(c, dim(M)) be the constant given by Lemma 3.10. Suppose that g is partially hyperbolic. We say that g is θ-accessible on D if, for each i = 1, …, c, there exists a continuous map

H^{i} : [0, 1] × D → N_{(m−2)ρ}(2D)

with t ↦ H^{i}(t, q) a 4-legged us-path for g originating at q, and, for some t_{0} ∈ (0, ρ/2), the condition

d(H^{i}(1, q), ζ_{t_{0}}^{i}(q)) < t_{0}θ  (7)

holds for all q ∈ D′. Here ζ^{i}, d, etc. are the structures described in the previous section, adapted at p.

The next lemma gives a criterion for central accessibility. A basic element of the proof is the “quadrilateral argument” of Brin, in which 4-legged paths are homotoped to the trivial path along 4-legged paths with endpoint in a fixed c-admissible disk. The reader unfamiliar with this argument may consult the survey paper [BuPuShWi] for a detailed description; the case c = 1 is also proved there. The case c > 1 is essentially proved in [ShWi], using an index argument.

Lemma 3.2 (Central accessibility criterion) Suppose β > 1/2. For every β′ ∈ (β, 1), there exist θ > 0, δ_{1} > 0 and ρ_{0} > 0 such that, for every c-admissible disk D of radius r(D) < ρ_{0}, if

• d_{C^{1}}(f, g) < δ_{1}, and

• g is θ-accessible on β′D,

then g is stably accessible on βD.

Proof of Lemma 3.2: Let β, β′ be given. Choose θ < (β′ − β)/4β′c. By continuity of the bundles E_{g}^{u}(p) and E_{g}^{c}(p) in (p, g), there exist δ_{1} > 0 and ρ_{0} > 0 such that, if d_{C^{1}}(f, g) < δ_{1}, if D is any c-admissible disk of radius r(D) = ρ ≤ ρ_{0}, and if s : [0, 1] → N_{(m−2)ρ}(D) is any 4-legged us-path for g with s(0), s(1) ∈ D, then

d(s(0), s(1)) ≤ ρ(β′ − β)/4c.  (8)

Let g, D be any such diffeomorphism and c-admissible disk. Suppose that g is θ-accessible on β′D. For i = 1, … c, we have maps H^{i} satisfying (7) with D′ = β′D. We show g is accessible on βD. Since the existence of such H^{i} is a C^{1}-open condition, this implies that g is stably accessible on βD.

By varying the lengths of the last 2 legs in the path t ↦ H^{i}(t, q), we may arrange that that H^{i}(1, q) ∈ D, for all q ∈ β′D. The reader may verify that it is possible to do so while preserving property (7). (If necessary, the value of θ can be reduced a little).

By a standard argument, the path t ↦ H^{i}(t, q), for q ∈ β′D, can be homotoped through 4-legged us-paths originating at q to the trivial path so that the endpoints stay in D during the homotopy. The trace of these endpoints along the homotopy gives a curve in D from q to H^{i}(1, q). More precisely, for i = 1, …, c, we obtain

Ψ^{i} : [0, 1] × [0, 1] × β′D → N_{θ}(D)

such that, for all s ∈ [0, 1], t ↦ Ψ^{i}(s, t, q) is a us-path for g with Ψ^{i}(s, 1, q) ∈ D, Ψ^{i}(0, t, q) = q and Ψ^{i}(1, t, q) = H^{i}(t, q). Thus, s ↦ Ψ^{i}(s, 1, q) =: Φ_{s}^{i}(q) is a curve in D, from q to H^{i}(1, q). Every point on this curve is the endpoint of a us-path originating at q.

By (8), we have, for q ∈ β′D:

diam(Φ^{i}([0, 1] × {q})) < ρ(β′ − β)/4c.  (9)

For q ∈ β′D, we then extend the definition of Φ_{s}^{i}(q) to values of s > 1 by the inductive formula

Φ_{t+m}^{i}(q) = Φ_{t}^{i}(Φ_{m}^{i}(q)),

for t ∈ (0, 1] and m ∈ N. How far Φ_{s}^{i}(q) can be extended in s depends of course on q. Note, however, that (7) gives, for m ∈ N,

d(Φ_{m}^{i}(q), ζ_{mt_{0}}^{i}(q)) = d(H^{i}(1, H^{i}(1, ⋯, H^{i}(1, q) ⋯)), ζ_{mt_{0}}^{i}(q))
= d(H^{i}(1, q_{m−1}), ζ_{t_{0}}^{i}(q′_{m−1}))
≤ d(H^{i}(1, q_{m−1}), ζ_{t_{0}}^{i}(q_{m−1})) + d(ζ_{t_{0}}^{i}(q_{m−1}), ζ_{t_{0}}^{i}(q′_{m−1}))
≤ t_{0}(β′ − β)/4β′c + d(q_{m−1}, q′_{m−1})
⋯

≤ mt_{0}(β′ − β)/4β′c.  (10)

As before, every point in the image of Φ_{·}^{i}(q) is the endpoint of a us-path originating at q, although this path can have more than 4 legs.

Let q_{0} = φ(−β′ρ/2(e_{1} + ⋯ + e_{c})), and define a map Z : [0, β′ρ]^{c} → M by:

Z(a_{1}, …, a_{c}) = ζ_{a_{1}}^{1} ⋯ ζ_{a_{c}}^{c}(q_{0}).

Note that Z is a homeomorphism onto β′D.

Next, consider the the map P : [0, ρβ′]^{c} → D defined by:

P (a_{1}, ⋯, a_{c}) = Φ_{a_{1}/t_{0}}^{1}Φ_{a_{2}/t_{0}}^{2} ⋯, Φ_{a_{c}/r_{1}t_{0}}^{c}(q_{0}).

Each point in the image of P is the endpoint of a us-path for g originating at q_{0}. We claim that D is in the interior of its image. Since Z is a homeomorphism onto β′D, it suffices to show that d_{C^{0}}(P, Z) < d(∂βD, ∂β′D) = ρ(β′ − β)/2.

If a = (a_{1}, …, a_{c}) ∈ [0, β′ρ]^{c}, with a_{i} = t_{0}(m_{i}+s_{i}), m_{i} ∈ N and s_{i} ∈ (0, 1], then, by (9), and (10),

d(P (a), Z(a)) = d(Φ_{a_{1}/t_{0}}^{1}Φ_{a_{2}/t_{0}}^{2} ⋯ Φ_{a_{c}/t_{0}}^{c}(q_{0}), ζ_{a_{1}}^{1}ζ_{a_{2}}^{2} ⋯ ζ_{a_{c}}^{c}(q_{0}))
≤ ∑_{i=1}^{c} d(Φ_{a_{i}/t_{0}}^{i}Φ_{a_{i+1}/t_{0}}^{i+1} ⋯ Φ_{a_{c}/t_{0}}^{c}(q_{0}), ζ_{a_{1}}^{i}Φ_{a_{i+1}/t_{0}}^{i+1} ⋯ Φ_{a_{c}/t_{0}}^{c}(q_{0}))
≤ ∑_{i=1}^{c} diam(Φ([0, 1] × {Φ_{a_{i+1}/t_{0}}^{i+1} ⋯ Φ_{a_{c}/t_{0}}^{c}(q_{0})}) +

∑_{i=1}^{c} d(Φ_{m_{i}}^{i}Φ_{a_{i+1}/t_{0}}^{i+1} ⋯ Φ_{a_{c}/t_{0}}^{c}(q_{0}), ζ_{m_{i}t_{0}}^{i}Φ_{a_{i+1}/t_{0}}^{i+1} ⋯ Φ_{a_{c}/t_{0}}^{c}(q_{0}))

≤ c(β − β′)/4c + ∑_{i=1}^{c} m_{i}t_{0}(β′ − β)/4β′c
< ρ(β′ − β)/2,

since t_{0}m_{i} < ρβ′. Then g is accessible on βD. ♢

3.2 Constructing the perturbation

Fix β′ ∈ (β, 1). The next lemma completes the proof of Lemma 1.1.

Lemma 3.3 For every δ, θ > 0, if r(D) is sufficiently small and R(D) is sufficiently large, then there exists g ∈ Diff_{μ}^{r}(M) such that

1. d_{C^{1}}(f, g) < δ,

2. d_{C^{0}}(f, g) < θ

3. g is θ-accessible on β′D, for each D ∈ D.

The proof of Lemma 1.1 now follows. Choose θ < σ satisfying the hypotheses of Lemma 3.2. Let g be given by Lemma 3.3. Since g is θ-accessible on β′D, g is stably accessible on D.♢ .

Proof of Lemma 3.3: Let δ, θ > 0 be given. We will perturb f by composing with a C^{∞}, volume preserving diffeomorphism ψ : M → M. We first estimate the effect of the composition ψ ∘ f on the partially hyperbolic splitting.

Say that ψ : M → M is supported on X ⊂ M if ψ = id outside of X ⊂ M. The next lemma states that if R(X) is sufficiently large, and p, q ∈ X are sufficiently close, then for any g, with gf^{−1} supported on X, the subspaces T ψ^{−1}(E_{g}^{u})(q) and E_{g}^{s}(q) are very close to Ẽ_{p}^{u}(q) and Ẽ_{p}^{s}(q), respectively.

Lemma 3.4 (Bundle Perturbation Lemma) There exists δ_{0} > 0 such that the following is true. For every γ > 0, there exists J > 0 such that, if ψ = g ∘ f^{−1} is supported on a set X with R(X) > J, and d_{C^{1}}(ψ, id) < δ_{0}, then for any p, q ∈ X with d(p, q) < J^{−1}, we have:

1. ^{}_{q̸}(E_{g}^{s}, Ẽ_{p}^{s}) ≤ γ, and

2. ^{}_{q̸}(T ψ^{−1}(E_{g}^{u}), Ẽ_{p}^{u}) ≤ γ.

Proof of Lemma 3.4: Let γ > 0 be given. Recall that the splittings T U_{p} = E^{u} ⊕ E^{c} ⊕ E^{s} and T U_{p} = Ẽ_{p}^{u} ⊕ Ẽ_{p}^{c} ⊕ Ẽ_{p}^{s} coincide at p.

By uniform continuity of the splitting E^{u} ⊕ E^{c} ⊕ E^{s}, uniformity of φ_{p}, and smoothness of ψ, there exists a continuous function θ_{1} : R_{+} → R_{+}, with θ_{1}(0) = 0, such that, for all p, q ∈ M,

q̸(T ψ(E^{u}), T ψ(Ẽ_{p}^{u})) ≤ θ_{1}(d(p, q))  (11)
q̸(E^{s}, Ẽ_{p}^{s}) ≤ θ_{1}(d(p, q)),  (12)

provided d_{C^{1}}(ψ, id) is small enough.

Let

λ = max_{p} (max (\frac{a_{p}}{b_{p}} ) , (\frac{B_{p}}{A_{p}} ))

and note that λ < 1, because f is partially hyperbolic. There exist C_{1}, θ_{0} > 0 such that, for all all subspaces F^{u}, F^{s} with

max{^{}(F^{u}, E^{u}), ^{}(F^{s}, E^{s})} ≤ θ_{0}

we have:

(T f^{−j}(F^{s}), T f^{−j}(E^{s})) ≤ C_{1}λ^{j}, and
(T f^{j}(F^{u}), T f^{j}(E^{u})) ≤ C_{1}λ^{j},

for all j ≥ 0. The splitting E_{g}^{u} ⊕ E_{g}^{c} ⊕ E_{g}^{s} depends continuously on g, and so

max{^{}(E_{g}^{u}, E^{u}), ^{}(E_{g}^{s}, E^{s})} ≤ θ_{0},

if d_{C^{1}}(ψ, id) (and so d_{C^{1}}(f, g)) is sufficiently small.

Fix positive R < R(X). If q ∈ X, then g^{i}(q) = f^{i}(q), for all i between 0 and R. For these q, we have

q̸(E_{g}^{s}, E^{s}) = ^{}_{q̸}(T g^{−R}E_{g}^{s}, T f^{−R}E^{s})
= ^{}_{q̸}(T f^{−R}E_{g}^{s}, T f^{−R}E^{s})
≤ C_{1}λ^{R}.

Similarly, for q ∈ X, g^{−i}(q) = f^{−i}ψ^{−1}(q) = f^{−i+1}g^{−1}(q), for all i between 1 and R − 1, and so

q̸(E_{g}^{u}, T ψ(E^{u})) = ^{}_{q̸}(T g(E_{g}^{u}), T g(E^{u}))
≤ C_{2}^{}_{g̸^{−1}(q)}(E_{g}^{u}, E^{u})
= C_{2}^{}_{g̸^{−1}(q)}(T f^{R−1}E_{g}^{u}, T f^{R−1}E^{u})
≤ C_{1}C_{2}λ^{R−1}.

Combining these inequalities with (11), we have shown: there exist λ, θ_{1} : R_{+} → R_{+}, and C > 0 such that, for any ψ sufficiently close to the identity and supported on X, for all R < R(X), and all q ∈ X, p ∈ M, we have

• ^{}_{q̸}(E_{g}^{s}, Ẽ_{p}^{s}) ≤ C(λ^{R} + θ_{1}(d(p, q))), and

• ^{}_{q̸}(T ψ^{−1}(E_{g}^{u}), Ẽ_{p}^{u}) ≤ 2^{}_{q̸}(E_{g}^{u}, T ψ(Ẽ_{p}^{u})) ≤ C(λ^{R} + θ_{1}(d(p, q))).

Hence if R is sufficiently large and d(p, q) is sufficiently small, these quantities are bounded by γ. ♢

We will also need the following lemma.

Lemma 3.5 There exists T > 0 such that, for ϵ > 0 sufficiently small, for any p ∈ M, and for any c-admissible disk D centered at p, there are C^{∞}, volume preserving flows ξ_{t}^{1}, …, ξ_{t}^{c} : U_{p} → U_{p} such that, for each i:

1. ξ_{t}^{i} = id outside N_{2ϵ}(D).

2. For q ∈ N_{ϵ}(D),

ξ_{t}^{i}(q) = ζ_{ϵt}^{i}(q).

(hence ξ_{t}^{i} preserves the leaves of W̃^{c} ∩ N_{ϵ}(D)),

3. d_{C^{1}}(id, ξ_{t}^{i}) < T |t|.

Proof of Lemma 3.5: Let G = φ_{p}^{−1}(D) = B^{c}(0, ρ), for some ρ > 0. Fix i, and let E be the divergence-free vector field on N_{2ϵ}(G) ⊂ R^{n} such that, for all v:

E(v) = ϵe_{i}.

Let ω be the Euclidean volume form on R^{n}, and let ϕ_{0} = i_{E}ω. Since E is divergence-free, the (n − 1)-form ϕ_{0} is closed: dϕ_{0} = di_{E}ω = div(E)ω = 0. Since N_{2ϵ}(G) is contractible, there exists an (n − 2)-form ν on N_{2ϵ}(G) such that dν = ϕ_{0}. We may choose ν so that

∥ν∥ ≤ 2ϵ^{2}, and ∥ν∥_{C^{1}} ≤ ϵ.

Let σ : N_{2ϵ}(G) → [0, 1] be a C^{∞} bump function, vanishing on a neighborhood of ∂N_{2ϵ}(G) and identically 1 on Nϵ(G), such that

∥dσ∥ ≤ 2/ϵ, and ∥dσ∥_{C^{1}} ≤ 2/ϵ^{2}.

Let ϕ = d(σν). Then

∥ϕ∥_{C^{1}} = ∥dσ ∧ ν + σϕ_{0}∥_{C^{1}}
≤ ∥dσ∥ · ∥ν∥_{C^{1}} + ∥dσ∥_{C^{1}} · ∥ν∥ + ∥σ∥_{C^{1}} · ∥ϕ_{0}∥_{C^{1}}
≤ 8

Hence ϕ has the following properties:

• ∥ϕ∥_{C^{1}} ≤ T, where T = 8,

• dϕ = 0,

• ϕ = ϕ_{0} on Nϵ(G),

• ϕ = 0 on ∂N_{2ϵ}(G).

Let X be the vector field on R^{n} satsifying i_{X}ω = φ, and let X_{t} be the flow generated by X. Let ξ_{t}^{i} = φ ∘ X_{t} ∘ φ^{−1}. Then ξ^{i} has the desired properties. ♢

Returning to the proof of Lemma 3.3, let T be given by Lemma 3.5. Let γ = θδ/100cT. Choose J > 0 according to Lemma 3.4.

Now suppose that D = {D_{1}, …, D_{k}} is any c-admissible family with R(D) > J and r(D) < J^{−1}. Choose η < r(D) so that the η-neighborhoods of any two c-admissible disks in D are disjoint.

To prove Lemma 3.3, it suffices to show that for any D ∈ D, there is a C^{∞} volume preserving diffeomorphism ψ, supported on the η-neighborhood N_{η}(D), such that

1. d_{C^{1}}(ψ, id) < δ,

2. d_{C^{0}}(ψ, id) < θ,

3. if \overline{f} is any diffeomorphism with f^{−1}\overline{f} supported on N_{η}(|D|) \ N_{η}(D), and d_{C^{1}}(\overline{f}, f) < δ, then ψ ∘ \overline{f} is θ-accessible on β′D.

To construct the final diffeomorphism g, we proceed disk by disk, constructing for each D_{i} ∈ D a diffeomorphism ψ_{i} suppported on N_{η}(D_{i}) so that ψ_{i} ∘ ψ_{i−1} ⋯ ψ_{i} ∘ f is θ- accessible on β′D_{i}. Then g = ψ_{k} ⋯ ψ_{1} ∘ f will satisfy the conclusions of Lemma 3.3.

Fix D ∈ D centered at p and choose ϵ < η/4c small enough to satisfy the hypotheses of Lemma 3.5. Let the flows ξ_{t}^{1}, …, ξ_{t}^{c} be given by Lemma 3.5.

For i = 1, …, c, let ϵ_{i} = 4iϵ, let Z_{i} = τ_{ϵ_{i}}^{u}(D), and let

N_{i} = N_{2ϵ}(Z_{i}) = τ_{ϵ_{i}}^{u}(N_{2ϵ}(D)).

The neighborhoods N_{1}, …, N_{c} are pairwise disjoint. Define ψ : M → M by

ψ(q) = \{ \begin{matrix}τ_{−ϵ_{i}}^{s}τ_{ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}ξ_{δ/T}^{i} τ_{−ϵ_{i}}^{u}(q) & if q ∈ N_{i}, for some i, \\ q & otherwise. \end{matrix}

Observe that ψ has the following properties:

• ψ preserves μ,

• ψ = id outside N_{1} ∪ ⋯ ∪ N_{c} ⊂ N_{η}(D), • ψ preserves the leaves of W̃^{c} outside of N_{1} ∪ ⋯ ∪ N_{c} and inside of

N_{ϵ}(Z_{i}), for i = 1, …, c.

• d_{C^{1}}(ψ, id) < δ.

Let \overline{f} be any diffeomorphism with d_{C^{1}}(\overline{f}, f) < δ and f^{−1}\overline{f} supported on N_{η}(|D|) \ N_{η}(D), and let g = ψ ∘ \overline{f}. It remains to show that g is θ-accessible on β′D.

We will now examine the behavior of the holonomy maps for g along us-paths whose corners are near the points:

τ_{ϵ_{i}}^{u}(p), τ_{ϵ_{i}}^{s}τ_{ϵ_{i}}^{u}(p), τ_{−ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}τ_{ϵ_{i}}^{u}(p).

In analogue with τ_{t}^{u} and τ_{t}^{s}, which translate along leaves of W̃^{u}, W̃^{s}, respectively, we introduce maps π_{t}^{u}, π_{t}^{s} : N_{η}(β′D) → B, which translate along W_{g}^{u} and W_{g}^{s} leaves:

{π_{t}^{u}(q)} = W_{g}^{u}(q) ∩ W̃^{cs}(τ_{t}^{u}(q))
{π_{t}^{s}(q)} = W_{g}^{s}(q) ∩ W̃^{cu}(τ_{t}^{s}(q)).

If d_{C^{1}}(f, g) is sufficiently small, these maps are well-defined for |t| ≤ ϵ_{c}. Notice that if we were to replace W_{g}^{u} and W_{g}^{s} with W̃^{u} and W̃^{s}, these equations would instead define τ_{t}^{u} and τ_{t}^{s}, respectively. Between W̃^{cs} leaves, π_{t}^{u} is the W^{u}-holonomy (and similarly for π_{t}^{s}). Lemma 3.4 will allow us to predict the behavior of these maps. The upshot is:

On the appropriate domains, π^{u} ∼ ψτ^{u} and π^{s} ∼ τ^{s},

where we will be precise about ∼ later.

For i = 1, … c, let

h^{i} = π_{−ϵ_{i}}^{s}π_{−ϵ_{i}}^{u}π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}.

Then h^{i} is a homeomorphism of N_{ϵ}(β′D) onto its image. Observe that h^{i}(q) is the endpoint of a 4-legged us-path for g, originating at q. By construction, these paths depend continuously on q, and so there are continuous maps

H^{i} : [0, 1] × β′D → N_{η}(D)

with t ↦ H^{i}(t, q) a 4-legged us-path for g, such that H^{i}(0, q) = q and H^{i}(1, q) = h^{i}(q). The rest of the argument goes as follows. We will show that π^{u} ∼ ψτ^{u} and π^{s} ∼ τ^{s}, which will imply that h^{i} ∼ τ_{−ϵ_{i}}^{s}ψτ_{−ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}. Since τ^{u} and τ^{s} are just translations, ψ = ζ_{t_{0}}^{i} on τ_{ϵ_{i}}(D), and ψ = id on τ_{−ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}τ_{ϵ_{i}}(D), we find that h^{i} ∼ ζ_{t_{0}}^{i}, where t_{0} = ϵδ/T. The remainder of the argument is devoted to making ∼ precise.

Lemma 3.6 (Holonomy Perturbation Lemma) For ϵ and δ sufficiently small,

d_{C^{0}}(h^{i}, ζ_{t_{0}}^{i}) ≤ t_{0}θ

where t_{0} = ϵδ/T, and the C^{0}-distance is measured on β′D.

Proof of Holonomy Perturbation Lemma 3.6: For δ sufficiently small, there exists a neighborhood Q ⊂ N_{ϵ}(β′D) of β′D such that, for i = 1, …, c,

π_{ϵ_{i}}^{u}(Q) ⊆ ψτ_{ϵ_{i}}^{u}(N_{ϵ}(β′D)),
π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(Q) ⊆ τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}(N_{ϵ}(β′D)), and
π_{−ϵ_{i}}^{u}π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(Q) ⊆ τ_{−ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}(N_{ϵ}(β′D)).

We now show that d(h^{i}(q), ζ_{t_{0}}^{i}(q)) ≤ t_{0}θ, for all q ∈ Q and i = 1 …, c, which implies the conclusion of the lemma.

From the definition of ψ, we write, for q ∈ Q,

ζ_{t_{0}}^{i}(q) = ξ_{t_{0}/ϵ}^{i}(q)
= ξ_{δ/T}^{i} (q)
= τ_{−ϵ_{i}}^{s}τ_{−ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}(q)

= τ_{−ϵ_{i}}^{s}ψτ_{−ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}(q),  (13)

the final equality a consequence of the fact that ψ is supported on N_{1}∪⋯∪N_{c}, which is disjoint from π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(Q). On the other hand,

h^{i}(q) = π_{−ϵ_{i}}^{s}π_{−ϵ_{i}}^{u}π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q).  (14)

We show that the corresponding factors in the two compositions (13) and (14) satisfy the desired inequality. More specifically, we show that, restricted to the appropriate domains, the distances d_{C^{0}}(π_{±ϵ_{i}}^{u}, ψτ_{±ϵ_{i}}^{u}) and d_{C^{0}}(π_{±ϵ_{i}}^{s}, τ_{±ϵ_{i}}^{s}) are bounded by θt_{0}/4.

First, consider the maps ψ ∘ τ_{ϵ_{i}}^{u} and π_{ϵ_{i}}^{u} on the domain Q. Recall that, restricted to W̃^{cs} leaves, τ_{ϵ_{i}}^{u} is the W̃^{u} holonomy map. But ψ ∘ τ_{ϵ_{i}}^{u} sends W̃^{cs} leaves in Q ⊂ N_{ϵ}(β′D) to W̃^{cs} leaves: τ_{ϵ_{i}}^{u} sends W̃^{c} leaves to W̃^{c} leaves, and ψ preserves W̃^{cs} leaves in τ_{ϵ_{i}}^{u}(Q) ⊂ N′_{i}. It follows that, restricted to W̃^{cs}(q)∩Q, the map ψ ∘ τ_{ϵ_{i}}^{u} is the ψ(W̃^{u})-holonomy map to the transversal W̃^{cs}(τ_{ϵ_{i}}^{u}(q)), where ψ(W̃^{u}) is the image of W̃^{u} under ψ. Recall that π_{ϵ_{i}}^{u} restricts to the W_{g}^{u}-holonomy map between W̃^{cs}(q) ∩ N_{ϵ}(β′D) and W̃^{cs}(τ_{ϵ_{i}}^{u}(q)).

Thus, between W̃^{cs} leaves, we are comparing the holonomy maps for the foliations W_{g}^{u} and ψ(W̃^{u}). To compare the holonomies for W_{g}^{u} and ψ(W̃^{u}), we first apply the smooth change of coordinates p ↦ ψ^{−1}(p) and compare the holonomies for ψ^{−1}(W_{g}^{u}) and W̃^{u}. Since d_{C^{1}}(ψ, id) is small, this change of coordinates distorts distances by a factor very close to 1.

The tangent distributions to ψ^{−1}(W_{g}^{u}) and W̃^{u} are T ψ^{−1}E_{g}^{u} and Ẽ^{u}, respectively. According to the Bundle Perturbation Lemma 3.4, the distributions T ψ^{−1}(E_{g}^{u}) and Ẽ^{u} are close; in particular,

q̸(T ψ^{−1}(E_{g}^{u}), Ẽ^{u}) ≤ γ = θδ/100cT,  (15)

for all q ∈ N_{η}(D). We now apply the next elementary lemma.

Lemma 3.7 Let F be a continuous foliation of B ⊂ U_{p} with C^{1}, u-dimensional leaves, transverse to Ẽ^{s} ⊕ Ẽ^{c}. Let T_{1} and T_{2} be smooth disks tangent to Ẽ^{s} ⊕ Ẽ^{c}. Assume that both the F- and W̃^{u}-holonomy maps between T_{1} and T_{2} are defined, and denote them by h^{F} and h_{W^{u}}, respectively. Then, for all q ∈ T_{1},

d(h^{F}(q), h_{W^{u}}(q)) ≤ dist(T_{1}, T_{2}) · sup_{q∈B}^{}_{q̸}(T F, Ẽ^{u}).

The analogous statement holds for s-dimensional foliations transverse to Ẽ^{s} ⊕ Ẽ^{c}.

Applying Lemma 3.7 to the foliation ψ^{−1}W_{g}^{u}, and using inequality (15), we obtain that, for any two transversals T_{1} ⊂ Q and T_{2} = τ_{ϵ_{i}}^{u}(T_{1}), and q ∈ T_{1},

d(h^{ψ^{−1}(W_{g}^{u})}(q), h_{W^{u}}(q)) ≤ dist(T_{1}, T_{2}) · sup_{q∈N_{η}(D)}^{}_{q̸}(T ψ^{−1}(E_{g}^{u}), Ẽ^{u})
≤ ϵ_{i}γ
≤ (4cϵ)γ
≤ (4cϵ)(θδ/100cT)
< θ(ϵδ/8T)
= θt_{0}/8.

But then, for all q ∈ Q,

d(π_{ϵ_{i}}^{u}(q), ψτ_{ϵ_{i}}^{u}(q)) = d(ψh^{ψ^{−1}(W_{g}^{u})}(q), ψh_{W^{u}}(q))
≤ Lip(ψ)θt_{0}/8
≤ θt_{0}/4.

Similarly, for q ∈ ψτ_{ϵ_{i}}^{u}(Q),

d(π_{ϵ_{i}}^{s}(q), τ_{ϵ_{i}}^{s}(q)) ≤ 2ϵ_{i}^{}(E_{g}^{s}, Ẽ^{s})
< θt_{0}/4.

Combining these inequalities and using the fact that τ_{ϵ_{i}}^{s} is an isometry, we have, for all q ∈ Q,

d(π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q), τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}(q)) ≤ d(π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q), τ_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q)) + d(τ_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q), τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}(q))
= d(π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q), τ_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q)) + d(π_{ϵ_{i}}^{u}(q), ψτ_{ϵ_{i}}^{u}(q))
< θt_{0}/4 + θt_{0}/4
= θt_{0}/2.

Proceeding in this fashion, we obtain that for q ∈ Q,

d(π_{−ϵ_{i}}^{s}π_{−ϵ_{i}}^{u}π_{ϵ_{i}}^{s}π_{ϵ_{i}}^{u}(q), τ_{−ϵ_{i}}^{s}ψτ_{−ϵ_{i}}^{u}τ_{ϵ_{i}}^{s}ψτ_{ϵ_{i}}^{u}(q)) < θt_{0},

which completes the proof. ♢

This completes the proof of Lemma 3.3. We have now shown that for each i, there exists

H^{i} : [0, 1] × β′D → N_{η}(D)

with t ↦ H^{i}(t, q) a 4-legged us-path for g, such that H^{i}(0, q) = q and

d(H^{i}(1, q), ζ_{t_{0}}^{i}(q)) = d(h^{i}(q), ζ_{t_{0}}^{i}(q)) < θt_{0},

where t_{0} = ϵδ/T. Hence, g is θ-accessible.♢ .

3.3 The symplectic case

If f preserves a symplectic form ω, then the perturbation g can also be made symplectic.

As in the proof for the volume preserving case, we begin with a local system of C^{∞} charts φ_{p} : B^{2m}(0, 1) → M, defined for each p ∈ M, where 2m = n. Similar to the volume preserving case, these charts can be chosen to have the following properties:

1. φ_{p}(0) = p,

2. T_{0}φ_{p} sends the splitting T_{0}R^{n} = R^{u} ⊕ R^{c} ⊕ R^{s} to the splitting T_{p}M = E^{u} ⊕ E^{c} ⊕ E^{s},

3. the symplectic form φ_{p}^{∗}ω is a linear pullback of the standard symplectic

form on R^{2m}:

φ_{p}^{∗}ω = A_{p}^{∗}(∑ dp_{i} ∧ dq_{i}),

for some linear map A_{p} on R^{2m},

4. p ↦ φ_{p} is a uniformly continuous map from M to C^{1}(B^{n}(0, 1), M). The dependence of φ_{p}, A_{p} on f is also continuous.

By Darboux’s theorem, for each p ∈ M, there exists a neighborhood U_{p} of p and coordinates κ_{p} : U_{p} → R^{2m} such that, in these coordinates, ω takes the standard form ∑ dp_{i} ∧ dq_{i}. For each p, T_{p}κ_{p} sends the splitting T_{p}M = E^{u}(p) ⊕ E^{c}(p) ⊕ E^{s}(p) to a splitting R^{2m} = R_{p}^{u} ⊕ R_{p}^{c} ⊕ R_{p}^{s}. Let A_{p} : R^{2m} → R^{2m} be a linear map that sends B^{2m}(0, 1) into κ_{p}(U_{p}) and sends the trivial splitting R^{2m} = R^{u} ⊕ R^{c} ⊕ R^{s} to R^{2m} = R_{p}^{u} ⊕ R_{p}^{c} ⊕ R_{p}^{s}, chosen to depend continuously on p, f. Then φ_{p} = κ_{p}^{−1} ∘ A_{p} satisfies properties 1.-4.

With this modification, the proof of the Main Theorem in the symplectic case proceeds exactly as in the volume preserving one, replacing “μ” by “ω”, until the proof on Lemma 3.3. Since we will modify slightly the statement and proof of this lemma, we restate it here in the symplectic case.

Lemma 3.8 For every δ, θ > 0, if r(D) is sufficiently small and R(D) is sufficiently large, then there exists g ∈ Diff_{ω}^{r}(M) such that

1. d_{C^{1}}(f, g) < δ,

2. d_{C^{0}}(f, g) < θ, and 3. each D ∈ D is covered by c-admissible disks βV_{1}, …, βV_{k} such that g

is θ-accessible on V_{i}, for each i.

Remark: If θ and δ are sufficiently small, then any g ∈ Diff_{ω}^{r}(M) satisfying conditions 1.- 3. in Lemma 3.8 is stably accessible on D; for then Lemma 3.2 implies that g is stably accessible on each βV_{i}, which implies stabe accessibility on their union, which contains D.

Proof of Lemma 3.8. Let D be a c-admissible family. Using Lemma 3.10 below, we will cover each D ∈ D with c-disks βV_{1}, …, βV_{k}. The lemma associates to each i an open set (a union of balls) N(D, i) ⊂ N_{η}(D); for different i, these sets are disjoint. We will then perturb inside of N(D, i) to obtain θ-accessibility on V_{i}.

Similar to the volume preserving case, we will need to show that if r(D) is sufficiently small and R(D) is sufficiently large, then for each D ∈ D and each c-disk V_{i} in the cover of D, there is a symplectic C^{∞} diffeomorphism ψ, supported on N(D, i), with

(a) d_{C^{1}}(ψ, id) < δ,

(b) d_{C^{0}}(ψ, id) < θ,

(c) if \overline{f} is any diffeomorphism with f^{−1}\overline{f} supported on N_{η}(|D|) \ N(D, i) and d_{C^{1}}(\overline{f}, f) < δ, then ψ ∘ \overline{f} is θ-accessible on V_{i}.

Each perturbation ψ is supported on a union of balls (as opposed to a tubular neighborhood); this allows for symplectic perturbations. The next lemma replaces Lemma 3.5 for the symplectic case.

Lemma 3.9 There exists T > 0 such that, for ϵ > 0 sufficiently small, for each p ∈ M and q ∈ B_{1/2}(p), there are C^{∞}, symplectic flows ξ_{t}^{i} = ξ_{t}^{i,q} : M → M, i = 1, …, c, such that, for each i:

1. ξ_{t}^{i} = id outside B_{2ϵ}(q)

2. For x ∈ B_{ϵ}(q),

ξ_{t}^{i}(x) = ζ_{ϵt}^{i}(x).

(hence ξ_{t}^{i} preserves the leaves of W̃^{c} ∩ B_{ϵ}(q)),

3. d_{C^{1}}(id, ξ_{t}^{i}) < T |t|.

Here, all balls B_{ρ}(q) are measured in the d_{p}-metric, and all other invariant structures W̃^{c}, ζ^{i}, etc. are adapted at p.

Proof of Lemma 3.9. Let p, q, i be given and let v = φ_{p}^{−1}(q) ∈ B^{n}(0, 1/2). We will explain how T is chosen later. Since constant vector fields are locally Hamiltonian with respect to φ_{p}^{∗}ω = A_{p}^{∗}(∑ dp_{i} ∧ dq_{i}), there exists a Hamiltonian vector field X^{i}, supported on B^{n}(v, 1/2), such that X^{i} = e_{i} on B^{n}(v, 1/4). Since the C^{1}-size of φ_{p} is uniformly controlled, there exists a T_{0} > 0, independent of p, q, i, such that ∥X^{i}∥_{C^{1}} < T_{0}. Given ϵ < 1/4, let

Y^{i}(x) = ϵX^{i}((x − v)/4ϵ).

Then Y^{i} is Hamiltonian (if X^{i} has Hamiltonian H, then Y^{i} has Hamiltonian 4ϵ^{2}H((x − v)/4ϵ)), is supported on B^{n}(v, 2ϵ), and satisfies ∥Y^{i}∥_{C^{1}} ≤ 4T_{0}. Furthermore, Y^{i} = ϵe_{i} on B^{n}(v, ϵ). The vector field (φ_{p})_{∗}Y^{i} generates the desired flow ξ_{i}. Clearly T can be chosen to depend only on T_{0} and other uniform data. ♢

Next, we choose the balls. The proof of the next lemma is an elementary exercise in Euclidean geometry.

Lemma 3.10 There exists m > 2, depending only on c and dim(M), such that, for ϵ > 0 sufficiently small and all p ∈ M, there exist k > 0 and points

{q_{i,j} | i = 1, …, c, j = 1, …, k} ⊂ N_{(m−2)ϵ}(D)

with the following properties:

1. there exist p_{1}, …, p_{k} ∈ D and ϵ_{i,j} > 0 such that τ_{−ϵ_{i,j}}^{u}(q_{i,j}) = p_{j},

2. the balls in the collection

{B_{2ϵ}(q_{i,j}), B_{2ϵ}(τ_{ϵ_{i,j}}^{s}(q_{i,j})), B_{2ϵ}(τ_{ϵ_{i,j}}^{u}τ_{ϵ_{i,j}}^{s}(q_{i,j})) | i = 1, …, c, j = 1, …, k}

are pairwise disjoint,

3. the balls

B_{βϵ}(p_{1}), …, B_{βϵ}(p_{k})

cover D.

Given δ and θ, let T be given by Lemma 3.9, let γ = θδ/100cT, and choose J according to Lemma 3.4. Let D be any c-admissible family with R(D) > J and r(D) < J^{−1}. Let D ∈ D with center p. Proceeding as in the proof of Lemma 3.3, choose ϵ < θ/4mc satisfying the hypotheses of Lemma 3.9, where m is given by Lemma 3.10. Fix 1 ≤ i ≤ c, and let the points {q_{i,j}}, p_{i} be given by Lemma 3.10. Let V_{i} = V_{ϵ}(p_{i}); by Lemma 3.10 the disks βV_{1}, …, βV_{k} cover D. Let

N(D, i) = ⋃{B_{2ϵ}(q_{i,j}), B_{2ϵ}(τ_{ϵ_{i,j}}^{s}(q_{i,j})), B_{2ϵ}(τ_{ϵ_{i,j}}^{u}τ_{ϵ_{i,j}}^{s}(q_{i,j})) | j = 1, … c}.

We show that properties (a)-(c) above are satisfied for this i.

In each ball B_{2ϵ}(q_{i,j}), let ξ_{t}^{i,j} = ξ_{t}^{i,q_{j}} be the flow given by Lemma 3.9, with q = q_{j}.

Define ψ : M → M by

ψ(q) = \{ \begin{matrix}ξ_{δ/T}^{i,j} & if q ∈ B_{2ϵ}(q_{i,j}), for some j, \\ q & otherwise. \end{matrix}

Then ψ has the following properties:

• ψ^{∗}ω = ω,

• ψ = id outside N(D, i),

• ψ preserves the leaves of W̃^{c} outside of ⋃_{j} B_{2ϵ}(q_{i,j}) and inside of ⋃_{j} B_{ϵ}(q_{i,j}),

• d_{C^{1}}(ψ, id) < δ.

Let \overline{f} be any diffeomorphism with f^{−1}\overline{f} supported on N_{η}(|D|) \ N(D, i), and let g = ψ ∘ \overline{f}. Let It remains to show that g is θ-accessible on V_{i}.

By the same argument as in the proof of the Main Theorem, we obtain that for each x ∈ V_{i}, there is a 4-legged us-path for g from x to a point y ∈ N_{(m−2)ϵ}(D) such that

d(y, ζ_{t_{0}}^{i}(x)) < θt_{0},

In other words, g is θ-accessible on V_{i}.♢

Acknowledgement. We thank Keith Burns, François Ledrappier, Charles Pugh and Michael Shub for many useful conversations. Some of the ideas in this paper grew out of an attempt of the second author with Burns, Pugh and Shub to prove a C^{2} version of the results here. This work is partially supported by the NSF.

References

[BV] Bonatti, C. and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–194.

[Ar] Arnaud, M.-C., The generic sympletic C^{1} diffeomorphisms of 4-dimensional sympletic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, Prépublications d’Orsay 2000-17.

[BonDi] Bonatti, C. and L. J. Diaz Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. 143 (1996), 357–396.

[Br] Brin, M., Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature, Func. Anal. Appl. 9 (1975), 9–19.

[BrPe] Brin, M. and Ya. Pesin, Partially hyperbolic dynamical systems, Math. USSR Izvestija 8 (1974), 177–218.

[BuPuWi] Burns, K., C. Pugh and A. Wilkinson, Stable ergodicity and Anosov flows, Topology 39 (2000), 149–159.

[BuPuShWi] Burns, K., C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity, preprint.

[BuWi1] Burns, K. and A. Wilkinson, Stable ergodicity of skew products, Ann. Sci. École Norm. Sup. 32 (1999), 859–889.

[BuWi2] Burns, K. and A. Wilkinson, Better center bunching. in preparation.

[HiPuSh] Hirsch, M., C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, 1977.

[Lo] Lobry, C., Controllability of nonlinear systems on compact manifolds, SIAM J. Control 12 (1974) 1–4.

[Mo] Moser, J., On volume elements on manifolds¡ Trans. AMS. 120 (1965) 285–294.

[NiTö] Niţică, V. and A. Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology 40 (2001) 259–278.

[PP] Parry, W. and M. Pollicott, Stability of mixing for toral extensions of hyperbolic systems. Tr. Mat. Inst. Steklova 216 (1997), Din. Sist. i Smezhnye Vopr., 354–363.

[PugSh1] Pugh, C. and M. Shub, Stable ergodicity and partial hyperbolicity, in International Conference on Dynamical Systems: Montevideo 1995, a Tribute to Ricardo Mañé Pitman Res. Notes in Math. 362 (F.Ledrappier et al, eds.) 182-187.

[PugSh2] Pugh, C. and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. of Complexity 13 (1997), 125–179.

[PugSh3] Pugh, C. and M. Shub, Stable ergodicity and julienne quasiconformality, J. Eur. Math. Soc. 2 (2000) 1–52.

[ShWi] Shub, M. and A. Wilkinson, Stably ergodic approximation: two examples, Ergod. Th. and Dyam. Syst. 20 (2000) 875– 893.

[Vi] Viana, M., Dynamics: a probabilistic and geometric perspective Proc. ICM–98, Documenta Math. Extra Vol. I (1998) 557–578.