Stable ergodicity of the time-one map of a geodesic flow

Amie Wilkinson Northwestern University

April 18, 1997

Introduction

Let M be a closed, connected Riemannian manifold with volume form ω. A C^{2}, volumepreserving diffeomorphism f : M → M is stably ergodic if there is a neighborhood U of f in Diff_{ω}^{2}(M), the space of C^{2}, volume-preserving diffeomorphisms of M, such that every g ∈ U is ergodic. Grayson, Pugh and Shub showed that if φ_{t} : T_{1}S → T_{1}S is the geodesic flow on the unit tangent bundle of a closed surface S of constant negative curvature, then the time-one map φ_{1} is stably ergodic ([5]). This is the first known example of a stably ergodic diffeomorphism that is not structurally stable. In this paper, we extend this result to variable negative curvature. More precisely, our Main Theorem states:

Main Theorem: If S is a closed, connected negatively-curved Riemannian surface, and if φ_{t} : T_{1}S → T_{1}S is the geodesic flow, then the time-one map φ_{1} is stably ergodic.

This paper is meant to serve as a model for proving stable ergodicity for diffeomorphisms of a nonalgebraic origin. These diffeomorphisms have local behavior that is uniformly controlled and qualitatively looks the same at every point, but their quantitative local behavior varies from point to point (subject to the qualitative constraints).

For a closed manifold M, let Diff^{k}(M) denote the space of C^{k} diffeomorphisms of M in the C^{k} topology. If M is equipped with a smooth volume form ω, then denote by Diff_{ω}^{k}(M) the subspace of Diff^{k}(M) consisting of those diffeomorphisms that preserve ω. How the ergodic diffeomorphisms “sit inside” the space Diff_{ω}^{k}(M) is a fundamental question, without a complete answer at this date.^{1} When M is a circle, ergodicity is determined by rotation number, and it is not hard to see that ergodic diffeomorphisms form a residual set in Diff_{ω}^{k}(M), (for any k) whereas non-ergodic diffeomorphisms are dense. On the other hand, when M is a surface, and k is sufficiently large, KAM theory guarantees that there are open sets of nonergodic diffeomorphisms, the nonergodicity manifested in a quite complicated dynamical picture. If M is a torus (for example) there are also the Anosov diffeomorphisms.

1The question for k = 0, i.e. for the space of homeomorphisms Homeo_{μ}(M), is settled by a classical result of Oxtoby and Ulam, which states that ergodic diffeomorphisms are residual.

Volume-preserving Anosov diffeomorphisms are stable in at least two senses of the word. Being uniformly hyperbolic, they are structurally stable. Since all C^{2} volumepreserving Anosov diffeomorphisms are ergodic ([1]) and the Anosov property is open in the C^{k} topology, Anosov diffeomorphisms are stably ergodic. On any manifold M supporting an Anosov diffeomorphism^{2} there is then an open family in Diff_{ω}^{2}(M) of ergodic diffeomorphisms, namely the Anosov diffeomorphisms.

Until recently, Anosov diffeomorphisms were the only class of diffeomorphisms known to be stably ergodic.

Question: Which manifolds support stably ergodic diffeomorphisms?

In [4], Brin and Pesin investigated the dynamical properties of diffeomorphisms like the time-one map of an Anosov flow, which they termed “partially hyperbolic.” They showed that such maps have certain transitivity properties that are preserved under perturbations. Extending some of the techniques in [4], Grayson, Pugh and Shub showed in [5] that if S is a closed surface of constant negative curvature and φ is its geodesic flow, then the time-t map φ_{t} is is stably ergodic. This provided the first example of a stably ergodic diffeomorphism that is not an Anosov diffeomorphism, and hence the first example of a stably ergodic map that is not also structurally stable.

The results of [5] readily generalize to other algebraic examples of partially hyperbolic diffeomorphisms on three-manifolds ^{3} They do not readily generalize to nonalgebraic examples. In this paper, we indicate how some of these obstacles can be overcome, focusing on the specific example of the geodesic flow for a surface of variable negative curvature.

We review some preliminary results and definitions in Section 1. In Section 2, we show that for a compact, negatively-curved surface S, the geodesic flow has the “stable/unstable accessibility” property described in [5]. We use the geometry of horospheres in the natural compactification of the universal cover S̃ to prove this.

In Section 3, we prove a regularity result about the strong stable and unstable foliations of the map φ_{1}. We show that these foliations are Hölder-continuous, with Hölder exponent arbitrarily close to 1. Further, we show that the Hölder constant of the foliation holonomy map is uniformly bounded in terms of the first two derivatives of φ_{1}. We use this to show that the Hölder continuity of the stable and unstable foliations is uniformly controlled under smooth perturbations of φ_{1}. The results in Section 3 are of independent interest and extend some of the results of Hirsch, Pugh and Shub in [9].

In Section 4, we use the results of the previous sections to prove the Main Theorem.

2All known examples are nilmanifolds or are finitely covered by nilmanifolds.

3See the footnote in Section 1.3.

0.1 Acknowledgment

The contents of this paper originally comprised a Ph.D. dissertation at the University of California, Berkeley. The author would like her advisor, Charles Pugh, for inspiration, guidance and support.

1 Preliminaries

1.1 Geodesic Flows

A Riemannian structure on a manifold S naturally gives rise to a flow on the tangent bundle T S. This flow is called the geodesic flow because its trajectories project to geodesics in S under the tangent bundle projection π : T S → S. We will consider the induced flow on the compact submanifold T_{1}S, the unit tangent bundle. While this flow is defined on the unit tangent bundle of S, for the sake of brevity we will at times call it the geodesic flow for S. For a construction of the geodesic flow see, for example, [13].

The geodesic flow φ : T_{1}S × R → R has several important properties. For any v ∈ T_{1}S, the unit-speed curve π ∘ φ_{t}(v) : R → S is a locally length-minimizing curve in S. The tangent bundle T S has a natural Riemannian metric (the “Sasake metric”), given by first decomposing the tangent space T_{v}T S at a point v ∈ T S into its horizontal and vertical components H_{v} and V_{v}. In this metric, the vertical subspace V_{v}, which is the kernel of T_{v}π, inherits the inner product from T_{πv}S via translation along v, and the horizontal space H_{v}, which is determined by the Riemannian connection, inherits the inner product from T_{πv}S via pullback under T_{v}π. The metric on T S is then given by the direct sum of these two forms on T (T S) = V ⊕ H. The geodesic flow preserves the Riemannian volume form ω given by this metric. Further, the flow preserves the trace of this metric on the submanifold T_{1}S. The measure m on T_{1}S induced by this metric is the Liouville measure. Locally, m is the product of the Riemannian measure on S with normalized Lebesgue measure on the fibers of T S.

The geodesic flow for a closed surface (or n-manifold) of negative sectional curvatures is ergodic with respect to m, and in fact Bernoulli. Ergodicity was proved first by Hedlund and Hopf in the constant curvature case (see [6] and [10]). In the same paper, Hopf also showed ergodicity of φ for a surface of finite area whose negative curvature and derivative of curvature are bounded, for instance, a closed negatively-curved surface. Anosov proved ergodicity in the general case case in [1], and Ornstein and Weiss proved in [15] that the flow is Bernoulli, meaning for every t the time - t diffeomorphism φ_{t} is conjugate to a Bernoulli shift. In particular, φ_{t} is ergodic. An alternate proof of this fact follows from the proof of Theorem 4.1.

1.2 Anosov Flows and Diffeomorphisms

A good reference for this section is [18]. A flow ψ_{t} : M → M is an Anosov flow if there are constants C > 0, λ < 1 and a splitting of the tangent bundle into T f-invariant subbundles, called an Anosov splitting:

T M = H^{u} ⊕ H^{c} ⊕ H^{s},

where H^{c} is spanned by the generating vector field ψ̇ and:

∥T ψ_{t}(v)∥ ≤ Cλ^{t}∥v∥ for all t ≥ 0, v ∈ H^{s}
∥T ψ_{−t}(v)∥ ≤ Cλ^{t}∥v∥ for all t ≥ 0, v ∈ H^{u}

The geodesic flow for a closed, negatively-curved manifold is an Anosov flow.

Theorem 1.1 Let S be a complete Riemannian n-manifold of bounded negative sectional curvatures −b^{2} < K < −a^{2} < 0, with a, b > 0. Let φ_{t} : T_{1}S → T_{1}S be the geodesic flow. Then there is an Anosov splitting

T (T_{1}S) = H^{u} ⊕ H^{c} ⊕ H^{s}

with dim(H^{u}) = dim(H^{s}) = n − 1 and

∥T φ_{t}(v)∥ ≤ \frac{b}{a}e^{−at}∥v∥ for all t ≥ 0, v ∈ H^{s}
∥T φ_{−t}(v)∥ ≤ \frac{b}{a}e^{−at}∥v∥ for all t ≥ 0, v ∈ H^{u},

with respect to the canonical Riemannian metric on T_{1}S (see Section 1.1).

A proof of Theorem 1.1 can be found in [8], [13], Chapter 3 or [2], Appendix 21. For any Anosov flow, the distributions H^{s}, H^{u}, H^{c}, H^{u} ⊕ H^{c}, H^{s} ⊕ H^{c} are uniquely

integrable and integrate to give foliations W^{u}, W^{s}, W^{c}, W^{cu}, W^{cs} respectively ([9]). The foliation W^{c} is the center foliation, whose leaves consist of orbits of φ. The foliations W^{u} and W^{s} are called the strong unstable and strong stable foliations, respectively, and the foliations W^{cu} and W^{cs} are called the center unstable and center stable foliations. The latter are also called the weak unstable and weak stable foliations, respectively. The leaves of these foliations are injectively immersed, C^{r} submanifolds of M (if f is C^{r}). For geodesic flows, the strong stable and unstable foliations are also called the horocyclic foliations. We will denote them by H^{u} and H^{s}. Some of their regularity properties are discussed in [8]. We discuss their geometric interpretation in Chapter 2.

For a closed, negatively-curved surface, the unit tangent bundle is three-dimensional, and it follows from, e.g. [18], Theorem 5.18, that the stable and unstable foliations are C^{1}. Hopf used this smoothness in a crucial way to prove ergodicity for these flows. Anosov showed in [1] that, while these foliations may fail to be smooth in higher dimensions, they do have the property of being absolutely continuous. He used this weaker property to prove ergodicity of the geodesic flow for any closed, negativelycurved manifold. This property will be used in our proof of the Main Theorem as well, so we review it here.

Let D_{0} and D_{1} be smooth, n − s-dimensional embedded disks transverse to the s-dimensional foliation F = {F (p)}_{p∈M}. Let points p ∈ D_{0} and p′ ∈ D_{1} and a path γ : [0, 1] → F (p) with γ(0) = p, γ(1) = p′ be given. Then there is a subdisk D′_{0} ⊂ D_{0} such that the holonomy h_{γ} : D′_{0} → D_{1} is a homeomorphism onto its image. The foliation F is absolutely continuous if for every such pair of transversals and every such (leafwise) path γ, the holonomy h_{γ} is absolutely continuous; i.e. it takes sets of measure zero in D′_{0} to sets of measure zero in D_{1}. (The measure in question is the Riemann-Lebesgue measure of dimension n − s). Anosov proved in [1] that if ψ is a C^{2}, volume-preserving Anosov flow then the foliations W^{u} and W^{s} are absolutely continuous. Further, he showed that for every holonomy h_{γ} : D′_{0} → D_{1}, there exists a continuous Jacobian J : D′_{0} → R such that for every measurable subset A ⊂ D′_{0},

∫_{A} J dm = m(h_{γ}(A)),

where m denotes Riemannian measure on D′_{0} or D_{1}. (A proof of this fact can be found in [16]).

Because φ_{t} restricted to a leaf of W^{u} is an expanding diffeomorphism, it follows from [17] that the leaves of W^{u} are diffeomorphic to R^{n−1} (as are the leaves of W^{s}). This implies that the holonomy h_{γ} is determined by γ(0) and γ(1).

1.3 Normal Hyperbolicity

Ignoring for a moment that φ_{1} comes from a flow, we could alternately describe it as a diffeomorphism f with a T f-invariant splitting T M = H^{u} ⊕ H^{c} ⊕ H^{s} such that H^{u} and H^{s} are expanded and contracted exponentially under T f and ∥T f|_{H^{c}}∥ = 1. For much of the proof of Theorem 4.1 this is all we need to know about f. What restricts the generality of our result is that this description of φ_{1} is not sufficient for the entire proof. The reason is that we cannot conclude from this information alone that the center distribution H^{c} is integrable. There is a center manifold theory for diffeomorphisms of this nature, and so we are assured the existence of local invariant center manifolds ([18], Theorem III.8). These manifolds are tangent to the center distribution at a single point, and are not in general unique. The assumption that ∥T f|_{H^{c}}∥ is close to 1 (which would be the case if f were a perturbation of φ_{1}), does not imply the existence of a smooth center manifold everywhere tangent to H^{c}.^{4}

Since we will use center manifolds to carry out the proof of our Main Theorem, we need to use more information about φ_{t}. What we would like to say is that φ_{t} is hyperbolic “modulo” the foliation H^{c}. The theory of normal hyperbolicity developed in [9] provides us with the tools.

Definition: (see [9], p. 116.) Let F be a continuous foliation whose leaves are C^{1}. A C^{r} diffeomorphism f : M → M is r-normally hyperbolic to F if f permutes the leaves

m(T_{p}f|_{E^{u}}) > ρ > ∥T_{p}f|_{E^{c}}∥ > m(T_{p}f|_{E^{c}}) > \frac{1}{ρ} > ∥T_{p}f|_{E^{s}}∥

4That is, there is a diffeomorphism f of a closed manifold M with (smooth) T f-invariant splitting T M = E^{u} ⊕ E^{c} ⊕ E^{s}, and a ρ > 1 such that for all p ∈ M, but the distribution E^{c} is not integrable (an example can be found by inspecting Smale’s construction of an Anosov diffeomorphism on a six-dimensional nilmanifold in [19]).

of F and there is a T f-invariant splitting:

T M = N^{u} ⊕ T F ⊕ N^{s}

such that:

inf m(T_{p}f|_{N^{u}}) > 1,  sup ∥T_{p}f|_{N^{s}}∥ < 1
inf m(T_{p}f|_{N^{u}})∥T_{p}f|_{F}∥^{−r} > 1, sup ∥T_{p}f|_{N^{s}}∥m(T_{p}f|_{F})^{−r} < 1,

where m(A) := ∥A^{−1}∥^{−1} denotes the conorm of the operator A. An Anosov diffeomorphism is normally-hyperbolic to the foliation by points F = {p}_{p∈M}. If ϕ is an Anosov flow, then ϕ_{1} is normally hyperbolic to its orbit foliation. Incidentally, the converse does not hold: there are diffeomorphisms normally hyperbolic with respect to a smooth, 1-dimensional foliation that are not conjugate to the time-one map of any Anosov flow. ^{5}

Recall that an ϵ-pseudo orbit for a diffeomorphism f is a bi-infinite sequence {p_{n}}_{n∈Z} of points in M such that d(f(p_{n}), p_{n+1}) ≤ ϵ, for all n ∈ Z. Suppose that the foliation F is given by a family of plaques P (see Chapter 3 for a definition). If f : M → M preserves the foliation F, then the pseudo-orbit {p_{n}} respects P if f(p_{n}) and p_{n+1} lie in a common plaque of P. We now generalize the notion of expansiveness to normallyhyperbolic diffeomorphisms.

Definition: ([9], p. 116) f : M → M is plaque expansive if there exists an ϵ > 0 with the following property. If {p_{n}} and {q_{n}} are ϵ-pseudo orbits which respect P and if d(p_{n}, q_{n}) ≤ ϵ for all n, then for each n, p_{n} and q_{n} lie in a common plaque.

Remark: If f is normally hyperbolic at F and F is a smooth foliation, then f is plaque expansive (see [9], p. 117). Hence if ψ_{t} is an Anosov flow, then ψ_{1} is plaque-expansive at its orbit foliation.

Theorem 1.2 (see [9], Theorem 7.1, p. 117) Let F_{0} be a continuous foliation of the closed manifold M whose leaves are C^{r} injectively immersed. Let f_{0} be r-normally hyperbolic to F_{0}. If f_{0} is plaque expansive, then the pair (f_{0}, F_{0}) is structurally stable. That is, there is a neighborhood U of f_{0} in Diff^{r}(M) such that, for every f ∈ U, there is a foliation F and a homeomorphism h of M such that

1. h carries each leaf of F_{0} to a leaf of F.

2. hf_{0}(L) = fh(L), for every L ∈ F_{0}.

3. f is r-normally hyperbolic and plaque expansive at F.

5For an example, one can take an automorphism of a compact quotient of the three-dimensional Heisenberg group with eigenvalues 1, λ > 1 and λ^{−1}. As a side note, these Heisenberg examples are in fact stably ergodic. We leave the proof of this fact as a lengthy exercise; the proof is almost identical to the proof in [5].

Theorem 1.2 implies structural stability for Anosov flows. Since the time-1 map of an Anosov flow is also normally hyperbolic and plaque-expansive with respect to the flow’s orbit foliation, we also obtain the following.

Corollary 1.3 (Center manifolds persist under perturbations.) Let ϕ be a C^{1} Anosov flow on the closed manifold M. Then there is a neighborhood U of ϕ_{1} in Diff^{1}(M) such that for every f ∈ U, there is a unique T f-invariant splitting T M = E^{u} ⊕ E^{c} ⊕ E^{s} which approximates the Anosov splitting for φ. Tangent to this splitting are unique f-invariant foliations W^{c}(f), W^{u}(f), and W^{s}(f). The foliation W^{c}(f) is homeomorphic to the orbit foliation for ϕ.

Strictly speaking, the existence of the foliations W^{u}(f) and W^{s}(f) is not a direct corollary to Theorem 1.2 but follows from the existence of an invariant splitting such that E^{u} is exponentially expanded, etc., plus the strong stable manifold theorem for invariant sets (see, e.g. [18], p. 79 ff.).

Since ϕ_{1} is normally hyperbolic (in fact, normally expanding) and plaque expansive at the weak-stable foliation (and ϕ_{1}^{−1} is normally hyperbolic and plaque expansive at W^{cu}) we also obtain:

Corollary 1.4 (Weak unstable manifolds persist under perturbations.) With ϕ as in Corollary 1.3, there is a neighborhood U′ of ϕ_{1} in Diff^{1}(M) such that for every f ∈ U′, there are unique f-invariant foliations W^{cs}(f) and W^{cu}(f) tangent to E^{u} ⊕ E^{c} and E^{s} ⊕ E^{c}, respectively.

In Section 3 we show that if f is C^{1}-close to φ_{1}, then the W^{u}(f)-holonomy maps between local W^{cs}(f)-leaves are uniformly Hölder continuous (and similarly for the W^{s}(f)- holonomy between local W^{cu}(f)-leaves). In Section 4, we show that these holonomies are uniformly absolutely continuous, and C^{1} and uniformly nearly isometric when restricted to W^{c}(f)-leaves.

2 Accessibility

2.1 A Three-Legged Lemma

Let S be a negatively-curved surface, with geodesic flow φ : T_{1}S × R → T_{1}S. As in the previous section, denote by H^{u}, H^{s} the strong unstable and stable foliations for φ_{t} and by H^{u} and H^{s} the corresponding line-fields. Since φ_{t} is a geodesic flow, the plane field H^{u} ⊕ H^{s} it is contact: there is a smooth one form τ such that H^{u} ⊕ H^{s} = ker (τ) and τ ∧ dτ is nonvanishing.^{6} Hence the plane field H^{u} ⊕ H^{s} is totally non-integrable: the set of paths originating at a point p and everwhere tangent to H^{u} ⊕ H^{s} will fill a whole neighborhood of p. In this section we examine a coarse version of this infinitessimal property. We show that the set of all paths everywhere tangent either to H^{u} or to H^{s} fill up T_{1}S, uniformly, and in a way that persists under perturbations of φ_{1}.

6τ is the canonical contact one-form pdq. See [11], p. 230.

Define a H^{u,s}-path to be a path ψ : [0, 1] → T_{1}S consisting of a finite number of consecutive arcs, or legs, each arc everywhere tangent to either H^{u} or H^{s}; that is, lying wholly in a leaf of H^{u} or H^{s}. Fix the convention that the first leg of a H^{u,s}-path lies in H^{s}. Similarly, for f close to φ_{1} in the C^{1} sense, define a W^{u,s}-path to consist of consecutive W^{u}(f), W^{s}(f) legs, where W^{u}(f), W^{s}(f) are defined in Corollary 1.3. If such a path has n legs, then call it an n-legged path.

Lemma 2.1 There is a neighborhood V of φ_{1} in Diff^{2}(T_{1}S) and a constant N > 0 such that for every f ∈ V and any two points v_{0}, v_{1} ∈ T_{1}S, there is a three-legged W^{u,s}(f)-path from v_{0} to v_{1} of length ≤ N.

Recently, Katok and Kononenko [12] have provided an alternate proof of a similar result, which they use to prove cocycle stability results for certain partially hyperbolic systems. They show that for a contact Anosov flow the stable and unstable foliations have this accessibility property and that this property is stable under smooth perturbations of the time-one map. Their proof has a different flavor and for completeness we describe it here.

The type of map studied in [12] is an Anosov flow φ that preserves a contact 1-form τ.^{7} This class includes all geodesic flows for negatively-curved manifolds. The definition of an H^{u,s} - path for a general Anosov flow with Anosov splitting T M = H^{u} ⊕ H^{c} ⊕ H^{s} is the same as in the specific case of a geodesic flow: it is a path consisting of finitely many smooth legs, each wholly tangent to either H^{u} or H^{s}. Since the time-one map φ_{1} is normally hyperbolic to the orbit foliation, it is normally structurally stable, and any sufficiently C^{1}-close diffeomorphism f will have stable, unstable and center manifolds, by Theorem 1.2, and we define an n-legged W^{u,s} - path for such a perturbation in the obvious way.

Proposition 2.2 ([12], ) Let φ : M → M be a contact Anosov flow on a closed manifold. Then there is a neighborhood V of φ_{1} in Diff^{2}(M) and an integer j > 0 such that for every ϵ > 0 and every f ∈ V there exists a δ > 0 such that for every p, q ∈ M with d(p, q) < δ, there exists a j-legged W^{u,s} - path from p to q of length less than ϵ.

Roughly, the proof of Proposition [12] is as follows. Because H^{u} ⊕ H^{s} is a contact distribution, any two points p, q ∈ M can be connected by a smooth path everywhere tangent to H^{u}⊕H^{s}. First take p and q lying in the same center leaf. Using the transverse local product structure of H^{cu} and H^{cs}, one can approximate this smooth path by an m-legged path, for some m, connecting p to a point q′, with d(q, q′) << d(p, q). Then the codimension 1 manifold

C(q′) := {x ∈ H_{loc}^{u}(y) | y ∈ H_{loc}^{s}(q′)}

intersects the center manifold H^{c}(p) in a point p′ distinct from p. This gives an m + 2legged path from p to p′. From here it is straightforward to construct a 1-parameter family of m + 2-legged paths connecting p to a neighborhood of p in H^{c}(p). Thus any point q in a neighborhood of p can be connected to p by an m + 4-legged path: first connect q to y ∈ H^{c}(p) ∩ C(q) by a 2-legged path and then connect y to p by a m + 2-legged path. This proves accessibility for the unperturbed map φ_{1}.

7This means that τ(φ̇) = 1, dτ(φ̇, ·) = 0, and τ ∧ (dτ)^{n−1} is nondegenerate.

For f C^{2}-close to φ_{1}, the foliations W^{a}(f) (a ∈ {u, s, c, cs, cu}) locally uniformly approximate the corresponding foliations for φ_{1}, and so for p and q on the same center leaf, there is an m + 4-legged path connecting p to a point q′, with d(q, q′) << d(p, q). Following the argument above, we can conclude there is a uniform neighborhood of p covered by m + 8-legged W^{u,s}-paths.□

2.2 Hadamard Manifolds, Horospheres, and Horoballs

The proof of Lemma 2.1 uses several properties of connected, simply-connected complete Riemannian manifolds of nonpositive sectional curvature (called Hadamard manifolds) which we recall here (see [3]). Many of the properties of the standard hyperbolic plane (i.e. the Poincaré disk model) carry over to Hadamard manifolds. Most notably, just as the Poincaré disk has a compactification as a ball in R^{2}, with boundary a circle (1-sphere), so does any Hadamard manifold X have a sphere at infinity. Say that two unit speed geodesics γ_{i} : R → X are asymptotic if there is a constant c such that d(γ_{1}(t), γ_{2}(t)) ≤ c, for all t ≥ 0. The sphere at infinity, X(∞) is the set of equivalence classes of geodesics under this relation. The compactification \overline{X} = X ∪ X(∞) is topologized as follows: first, given x ∈ X, z_{1}, z_{2} ∈ \overline{X}, let

x̸(z_{1}, z_{2}) = ^{}(γ̇_{1}(0), γ̇_{2}(0)),

where γ_{i} is a geodesic from x to z_{i}. For x ∈ X, z ∈ X(∞), ϵ > 0, let

C_{x}(z, ϵ) = \{y ∈ \overline{X} | y ≠ x and ^{}_{x̸}(z, y) < ϵ\}.

The “cone topology” on \overline{X} is generated by the topology on X and the cones \{C_{x}(z, ϵ)\}. In the cone topology X(∞) is homeomorphic to a codimension-1 sphere.

In the hyperbolic plane H (of constant curvature −1) there are distinguished curves of constant geodesic curvature 1 called horospheres. In the Poincaré disk model, horospheres are represented by circles tangent to the boundary of the disk. For every point on the sphere at infinity there is a unique family of horospheres tangent to that point and every geodesic asymptotic to that point crosses every horosphere tangent to that point orthogonally.

If φ̃ : R × T_{1}H → T_{1}H is the geodesic flow for H then for v ∈ T_{1}H the leaves H̃^{s}(v), H̃^{u}(v) of the strong stable and unstable foliations for φ̃ project to horospheres in H through the basepoint of v, one tangent to the forward asymptote of v, and the other to the negative asymptote (see Figure 1). An analogous picture holds for any Hadamard manifold. For those manifolds with bounded negative curvature, horospheres are the traces in S̃ of the strong stable and unstable manifolds for the (lifted) geodesic flow.

Figure 1: Strong stable and unstable leaves project to horospheres.

We proceed to definte horospheres for a Hadamard manifold X. Let γ : R → X be a unit-speed geodesic with γ(0) = x. Define a function h_{t,γ} : X → R as follows:

h_{t,γ}(y) = d(y, γ(t)) − t.

A function g on a Riemannian manifold X is (strictly) convex if for every geodesic γ : R → X, the function g ∘ γ is (strictly) convex.

Lemma 2.3 The function h_{t,γ} converges to a finite limit h_{γ} as t → +∞, and this limit is a C^{1}, convex function.

For a proof of Lemma 2.3 see ([2], Appendix 21), or ([3], Lemma 3.4). The limiting function h_{γ} is called a Busemann function or a horofunction. In fact, Busemann functions for nonpositively curved manifolds are C^{2} (for a proof, see [7], Prop. 3.1). A horosphere is a level set of the Busemann function h_{γ}.

Note that if γ(0) = x, then x ∈ h_{γ}^{−1}(0). Each horosphere h_{γ}^{−1}(c) bounds a convex region called a horoball, h_{γ}^{−1}(−∞, c]. Each horosphere(/function/ball) has a unique center z ∈ X(∞) such that every geodesic that crosses the horosphere orthogonally is asymptotic to its center, and vice versa. When the curvature of X is strictly negative and bounded away from 0 and ∞, the unit-speed geodesics orthogonal to a given horosphere converge to each other uniformly exponentially in forward time and diverge exponentially in backward time. It follows that the strong stable and unstable foliations of the geodesic flow on the unit tangent bundle T_{1}X, project to foliations of X by horospheres. More precisely, a unit tangent vector v ∈ T_{1}X is tangent to a unique directed unit-speed geodesic, φ̃_{t}(v). The horospheres corresponding to the geodesics φ̃_{t}(v) and φ̃_{−t}(v) are called, respectively, the positive and negative horospheres through v, and denoted H^{+}(v) and H^{−}(v). The set of unit vectors orthogonal to H^{+}(v) and pointing toward the center (resp. pointing away from the center of H^{−}(v)) forms the leaf through v of the strong stable (resp. unstable) foliation for the geodesic flow on T_{1}X

When the curvature of X is bounded away from zero, the horospheres in X have the following additional properties:

1. For every two different points z_{0}, z_{1} ∈ X(∞) on the sphere at infinity, there is a unique geodesic γ : R ⟶ X with γ(−∞) = z_{0} and γ(∞) = z_{1}.

2. Two horoballs with different centers intersect in a bounded region in X.

3. Given a horoball B and a horofunction h with a different center than B, there exists an a ∈ R such that h^{−1}(−∞, a] ∩ B = ∅.

4. Busemann functions (and hence, horoballs) are strictly convex. If a geodesic intersects a horosphere, it does so either tangentially, orthogonally, or transversely, in exactly two points.

Properties 1 − 3 are equivalent for Hadamard manifolds (for a proof, see, [3]). Property 4 is stronger. It follows, e.g., from Lemma 4.3 in [7].

Properties 1 − 4 imply that horospheres behave nicely with respect to each other.

Lemma 2.4 Let X be a two-dimensional Hadamard space of bounded negative curvature −b^{2} < K < −a^{2}. If H_{1} is a horosphere and B_{2} is a horoball in X, then

1. H_{1} ∩ B_{2} is path-connected.

2. H_{1} ∩ int(B_{2}) is path-connected.

Corollary 2.5 With X as above, two distinct, nondisjoint horospheres in X intersect tangentially in a single point or transversely in exactly two points.

Remarks:

1. Lemma 2.4 does not follow from strict convexity alone, as two strictly convex planar curves can intersect in arbitrarily many points.

2. In general, although it is not proved here, two horospheres in an n-dimensional Hadamard manifold of strictly negative sectional curvatures intersect in a point or in a topological n − 2-sphere.

Denote by B_{1} the horoball bounded by H_{1} and by H_{2} the horosphere bounding B_{2}. The lemma holds trivially if H_{1} and H_{2} have the same center, so assume that their centers are distinct.

If H_{1} and H_{2} meet tangentially, then the interiors of B_{1} and B_{2} must be disjoint, since an outward normal direction uniquely determines a horosphere. From the convexity of B_{1} ∩ B_{2}, we must have card(H_{1} ∩ H_{2}) = 1. In this case (1) and (2) follow immediately.

Figure 2: H_{1} leaves and then re-enters B_{2}.

This leaves the case where H_{1} and H_{2} do not meet tangentially but cross transversely. Since B_{1} ∩ B_{2} is bounded and convex, H_{1} ∩ H_{2} must consist of a finite, even number of points and (1) and (2) are equivalent. If H_{1} ∩ B_{2} is not path-connected then, as a parametrized simple curve, H_{1} must leave and then re-enter B_{2}. We may assume that H_{1} leaves B_{2} at p and re-enters B_{2} for the first time, to the right of p at q, as shown in Figure 2. We have not yet specified whether int(B_{2}) lies to the left or right of H_{1}. If int(B_{1}) lay to the left of H_{1}, then p and q could not be connected by a path in B_{1} ∩ B_{2}, violating the convexity of B_{1} ∩ B_{2}.

This leaves the possibility that int(B_{1}) lies to the right of H_{1}, as shown in Figure 3. Let h be the Busemann function that determines H_{1}, so that B_{1} = h^{−1}(−∞, 0]. By property 3 above, there exists an a < 0 such that h^{−1}(−∞, a] ∩ B_{2} = ∅. This, together with the smoothness of h implies that there is a b < 0 and a point r lying on the arc of H_{2} between p and q such that h^{−1}(b) is tangent to H_{2} at r and h^{−1}(−∞, b] ∩ B_{2} ≠ ∅. Since H_{1} and H_{2} have different centers, this gives a contradiction. □

2.3 Proof of Lemma 2.1

Let S̃ be the universal cover of S equipped with the pulled-back Riemannian metric of S under the canonical projection p : S̃ → S. The Hadamard manifold S̃ has strictly negative curvature bounded, by compactness of S, between two constants: −b^{2} < K < −a^{2}.

Choose ṽ_{0}, ṽ_{1}, lifts of v_{0}, v_{1} under T p : T_{1}S̃ → T_{1}S so that the directed geodesics in S̃ tangent to ṽ_{0}, ṽ_{1} are not forward - asymptotic. This is possible because S is compact and so the limit set of the fundamental group of S acting on \overline{S̃} is the entire sphere at

Figure 3: int(B_{1}) lies to the right of H_{1}.

infinity (see [13], Lemma 3.9.7, p. 370). Let

h_{v_{i}}(x) = lim_{t→∞} d(x, π(φ̃_{t}(ṽ_{i}))) − t

be the Busemann function associated with the vector ṽ_{i}. By a similar argument, we may assume that the horospheres H_{i} := h−1 ; ṽ_{i}(0) (i = 0, 1) are disjoint (although this is not necessary for the argument, it simplifies the pictures). Let c = d(H_{0}, H_{1}) and let z_{i} ∈ S̃(∞) be the center of the horosphere H_{i}.

For each x ∈ H_{0}, there is a unique unit-speed geodesic γ_{x} such that γ_{x}(0) = x and γ_{x}(−∞) = z_{0}. The horosphere h_{γ_{x}}^{−1}(0) is tangent to the horosphere H_{0}, since the geodesic γ_{x} crosses H_{0} orthogonally.

Define Δ, a continuous real-valued function on the horosphere H_{0} as follows:

Δ(x) = inf_{y∈H_{1}} h_{γ_{x}}(y).

Because h_{γ_{x}} is a strictly convex function and H_{1} bounds a strictly convex set, for a given x the infimum above is attained by a unique y ∈ H_{1}.

The function Δ measures the (signed) distance between the horospheres H_{x} := h_{γ_{x}}^{−1}(0) and H_{1}. The distance between two horospheres is achieved along the geodesic connecting their centers (which intersects both horospheres orthogonally). If the interiors of the horoballs corresponding to H_{x} := h_{γ_{x}}^{−1}(0) and H_{1} are disjoint, then Δ is nonnegative. Otherwise, Δ is negative.

We are interested in Δ for the following reason: if Δ(x) = 0, then, by Lemma 2.4 there exists a unique y ∈ H_{1} such that h_{γ_{x}}(y) = 0, or in other words, the horosphere h_{γ_{x}}^{−1}(0) is tangent to both H_{0} and H_{1}. This tangent sphere provides a bridge between H_{0} and H_{1} from which we can construct an H^{u,s} path in T_{1}S̃ from ṽ_{0} to ṽ_{1}.

Figure 4: Δ(x_{0}) ≤ 0.

The horosphere H_{0} is a one-dimensional subspace of S̃, homeomorphic to R, so to show that Δ has a zero, it suffices to find x_{0}, x_{1} ∈ H_{0} such that Δ(x_{0}) ≤ 0 and Δ(x_{1}) ≥ 0.

Let γ_{0} be the unique unit-speed geodesic in S̃ such that γ_{0}(∞) = z_{0} and γ_{0}(−∞) = z_{1}. Let x_{0} and y_{0} be the (unique) intersections of the image of γ_{0} with H_{0} and H_{1}, respectively. Then, for all x sufficiently close to x_{0}, the horoballs corresponding to H_{1} and H_{x} intersect, and Δ(x) ≤ 0. (In fact, Δ(x_{0}) = −∞). (See Figure 4).

To find x_{1}, we first find a geodesic γ_{1} with γ_{1}(−∞) = z_{1} that intersects H_{0} in 2 points. Up to parametrization, γ_{0} is the unique geodesic from z_{1} that intersects H_{0} orthogonally. Pick y_{1} ∈ H_{1} near y_{0} and let γ_{1} be the unit-speed geodesic with γ_{1}(−∞) = z_{1} and γ_{1}(0) = y_{1}. Since horospheres and geodesics are smooth, if we pick y_{1} sufficiently close to y_{0}, the intersection of γ_{1} and H_{0} will remain transverse, although no longer orthogonal. By property (4) above, this means that γ_{1} intersects H_{1} in exactly 2 points. Let z_{2} = γ_{1}(∞). Let γ_{2} be a geodesic with γ_{2}(−∞) = z_{2} and γ_{2}(∞) = z_{0} (See Figure 5). Let x_{1} be the intersection of the image of γ_{2} with H_{0}. We claim that Δ(x_{1}) ≥ 0.

Since H_{x_{1}} is tangent to H_{0}, the interiors of their associated horoballs are disjoint. The geodesic γ_{1} crosses the horosphere H_{x_{1}} exactly once. Hence γ_{1} crosses H_{x_{1}} after crossing H_{0}. This means that Δ(x_{1}) ≥ d(H_{0}, H_{1}) = c > 0.

Hence there exists a horosphere H_{2} tangent to both H_{0} and H_{1}. Note that the stable leaves H^{s}(ṽ_{0}), H^{s}(ṽ_{1}) in T_{1}S̃ project under π to H_{0}, H_{1} respectively. Let x = H_{0} ∩ H_{2} and let y = H_{1} ∩ H_{2}. Because the intersections are tangential, they lift to intersections in T_{1}S̃: if H^{u} is the leaf of H^{u} in M̃ that projects to H_{2}, then π(H^{u} ∩ H^{s}(ṽ_{0})) = x and π(H^{u} ∩ H^{s}(ṽ_{1})) = y. Thus there is an H^{u,s} path ψ̃ in S̃ defined by traversing first H^{s}(ṽ_{0}), then H^{u}, and finally H^{s}(ṽ_{1}). (See Figure 6).

Figure 5: Δ(x_{1}) ≥ 0.

Figure 6: A three-legged H^{u,s} path in T_{1}S̃.

The path ψ̃ projects to a three-legged H^{u,s}-path in T_{1}S. Hence any two points in T_{1}S can be connected by some three-legged H^{u,s}-path.

The foliations H^{s} and H^{u} are at least C^{1}. Let ξ^{s} and ξ^{u} be unit stable and unstable vector fields tangent to the bundles H^{u} and H^{s} for the flow φ_{t}. These vector fields integrate to give C^{1} flows ψ^{+} and ψ^{−} on T_{1}S. The trajectories of ψ^{+} and ψ^{−} lie along the leaves of the strong stable and unstable foliations H^{s} and H^{u}, and are called the positive and negative horocyclic flows, respectively. (For a discussion of these flows and their ergodic properties, see [14]). We have shown that there exist t_{1}, t_{2}, t_{3} ∈ R, with t_{1} > 0 such that ṽ_{1} = ψ̃_{t_{3}}^{+}∘ψ̃_{t_{2}}^{−}∘ψ̃_{t_{1}}^{+}(ṽ_{0}), where ψ̃^{±} are the lifted positive and negative horocyclic flows on T_{1}S̃. These flows are smooth and hence their composition varies continuously in the parameter (t_{1}, t_{2}, t_{3}). We show now that the map Γ̃(s_{1}, s_{2}, s_{3}) = ψ̃_{s_{3}}^{+} ∘ψ̃_{s_{2}}^{−} ∘ψ̃_{s_{1}}^{+}(ṽ_{0}) is injective in a neighborhood of (t_{1}, t_{2}, t_{3}) in the positive half-space t_{1} > 0 in R^{3}. It follows from invariance of domain that Γ̃ is a local homeomorphism.

Let x, y, ṽ_{0}, ṽ_{1} be as above, with x lying to the left of ṽ_{0} and y lying to the right of ṽ_{1} and x, y belonging to the same horosphere H. Suppose that there exist x′, y′ near x, y in H_{0}, H_{1} and belonging to the horosphere H′. Assume that H and H′ meet H_{0}, H_{1} tangentially at these points.

Since the horocyclic foliations are orientable, the only possible configuration of x, y, x′, y′ (up to orientation) is shown in Figure 7. In theis configuration, x′ lies to the left of x on H_{0} and y′ lies to the left of y in H_{1} (this also follows from the fact that the horocyclic flows are orientation-preserving). Let B and B′ be the horoballs corresponding to the horospheres H and H′. The points x and y must lie outside of the the horoball B′ since H′ intersects H_{0} and H_{1} each in exactly one point. Similarly x′ and y′ lie outside of B. It follows that the infinite segment of H to the right of x intersects the arc of H′ between x′ to y′ in a point w_{1}. Likewise, the segment of H′ to the left of y′ intersects the segment of H between x and y in a point w_{2}. Finally, there is a third point of intersection w_{3}, where the H segment from x to y meets the H′ segment from x′ to y′. The three points w_{1}, w_{2}, w_{3} are distinct, since H and H′ are simple curves. This contradicts Corollary 2.5. This implies that Γ̃ is a local homeomorphism.

In Section 3 we show that under C^{2} small perturbations of φ_{1}, the strong stable and unstable foliations persist and are transversally Hölder-continuous. The leaves of these foliations are unique integral curves of new vector fields, ξ′_{1} and ξ′_{2}, which are C^{0} perturbations of ξ_{1} ξ_{2}. The perturbed vector fields ξ′_{1} and ξ′_{2} generate flows ψ^{+′} and ψ^{−′} that C^{0}-approximate the flows ψ^{+} and ψ^{−} uniformly on compact time intervals.

Fix v ∈ T_{1}S. Let Γ_{v} : R^{3} → T_{1}S be defined by:

Γ_{v}(t_{1}, t_{2}, t_{3}) := ψ_{t_{3}}^{+} ∘ ψ_{t_{2}}^{−} ∘ ψ_{t_{1}}^{+}(v).

Since the flows ψ^{±} lift to the flows ψ̃^{±}, the above arguments imply that, restricted to the half-space D = {(t_{1}, t_{2}, t_{3}) ∈ R^{3} | t_{1} > 0}, Γ_{v} is surjective and a local homeomorphism. Thus for every w ∈ T_{1}S, there exist constants r, ρ > 0 and a point s = (s_{1}, s_{2}, s_{3}) ∈ D such that Γ_{v}(s_{1}, s_{2}, s_{3}) = w and

index (Γ_{v}|_{∂B_{r}^{3}(s)}) = ±1,

Figure 7: A possible configuration of x, y, x′, y′.

with respect to every w′ ∈ B_{ρ}(w). The balls {B_{ρ}(w)}_{w∈T_{1}S} cover T_{1}S. Pass to a finite subcover {B_{ρ_{1}}(w_{1}), B_{ρ_{2}}(w_{2}), …, B_{ρ_{k}}(w_{k})} and let r_{i} and s^{i} ∈ D be such that

index (Γ_{v}|_{∂B_{r_{i}}^{3}(s^{i})}) = ±1,

with respect to every w′ ∈ B_{ρ}(w_{i}). Note that there exists an R > 0 such that for every i ≤ k, |s^{i}| ≤ R.

Since the property of having index ±1 is robust and Γ_{v} is continuous as a function of v, there is a neighborhood B_{ρ′}(v) of v such that for every v′ ∈ B_{ρ′}(v),

index (Γ_{v′}|_{∂B_{r_{i}}^{3}(s^{i})}) = ±1,

with respect to every w′ ∈ B_{ρ}(w_{i}). For every v ∈ T_{1}S there is such an open neighborhood B_{ρ′}(v), where of course ρ′ depends on v.

The collection of open sets {B_{ρ′}(v)}_{v∈T_{1}S} covers T_{1}S. Again, pass to a finite subcover by balls {B_{ρ′_{1}}(v_{1}), B_{ρ′_{2}}(v_{2}), …, B_{ρ′_{m}}(v_{m})}. Re-indexing, we now have constants N, r, ρ > 0 and sets {v_{1}, …, v_{m}}, {w_{i,1}, … w_{i,k_{i}}}_{i=1,k} such that for every 1 ≤ i ≤ m and every 1 ≤ j ≤ k_{i}, there is a point s^{i,j} ∈ D, with |s^{i,j}| ≤ N such that Γ_{i}(s^{i,j}) := Γ_{v_{i}}(s^{i,j}) = w_{i,j} and

index (Γ_{i}|_{∂B_{r}^{3}(s^{i,j})}) = ±1,

with respect to every w′ ∈ B_{ρ}(w_{i,j}).

For 1 ≤ i ≤ m, and for f sufficiently close to φ_{1}, there is a map Γ′_{i} : R^{3} → T_{1}S defined by:

Γ′_{i}(t_{1}, t_{2}, t_{3}) := ψ_{t_{3}}^{+′} ∘ ψ_{t_{2}}^{−′} ∘ ψ_{t_{1}}^{+′}(v_{i}),

where ψ^{±′} are the positive and negative W^{u,s}(f)- horocyclic flows. Then Γ′_{i} uniformly approximates Γ_{i} on the set [−N, N]^{3} ⊂ R^{3}. It follows that for f sufficiently close to φ_{1},

index (Γ′_{i}|_{∂B_{r}^{3}(s^{i,j})}) = ±1,

with respect to every w′ ∈ B_{ρ}(w^{i,j}). The result follows.□

Remark: The proof of Lemma 2.1 generalizes to noncompact manifolds of bounded negative curvature, provided that M has at least two closed geodesics (equivalently, π_{1}(M) does not have a global fixed point at infinity).

If π_{1}(M) has a global fixed point at infinity, then Lemma 2.1 does not hold. In particular, if two points in T_{1}H lie in the same leaf of the weak-stable foliation, then there is no three-legged H^{u,s} in T_{1}H path connecting them.

3 Uniform Hölder-Continuity

If f is C^{1} close to φ_{1}, then the estimates in Theorem 1.1 will hold; that is, we will have that ∥T f|_{E^{u}}∥ and ∥T f|_{E^{s}}∥ are bounded above and below by a function involving the bounds on the curvature −b^{2} < −a^{2}. This information, combined with standard invariant manifold techniques (in, for example, [18]) implies that the splitting E^{u} ⊕ E^{c} ⊕ E^{s} is α-Hölder continuous, with α determined by the ratio \frac{a}{b}. This is not enough for our purposes, unless the curvature is pinched close to one (i.e. a/b is close to 1). We would like to exploit the volume-preserving aspect of φ and use the fact that pointwise the expansion and contraction rates of T f|_{E^{u}} and T f|_{E^{s}} are nearly reciprocal to each other.

In this section, we prove some regularity results about the strong stable and unstable foliations and the invariant splitting of a C^{2} perturbation of the time-one map φ_{1}. Recall that a function s : X → Y between compact metric spaces is said to be Höldercontinuous if there exist constants 0 < α ≤ 1 and H > 0 such that:

d_{Y} (s(x), s(x′)) ≤ H · d_{X}(x, x′)^{α},

for all x, x′ ∈ X. In the expression above, α is called the Hölder exponent and H the Hölder constant.

3.1 Hölder-Continuity of the Strong Unstable Foliation.

Definition: Let M be a closed manifold. A C^{r} pre-lamination of M is a continuous choice of C^{r}-embedded disk W (p) through each p ∈ M. The choice of W (p) is given by a covering of M by plaque charts (U_{j}, σ_{j}), such that:

• σ_{j} : U_{j} → Emb^{r}(D^{k}, M) is a continuous section of the bundle of C^{r} embeddings of the k-disk D^{k} = (−1, 1)^{k} into M.

• For p ∈ U_{j}, σ_{j}(p)(0) = p and σ_{j}(p)(D^{k}) = W (p).

Let F := {F (p)}_{p∈M} be a foliation of M and let T = {T (p)}_{p∈M} be a pre-lamination of M by C^{1} disks transverse to F and of complementary dimension. Recall that if γ : [0, 1] → F (p) is a leafwise path with γ(0) = p and γ(1) = q, then on a sufficiently small subdisk T (p)′ ⊂ T (p) there is a well-defined holonomy map

h_{γ} : T (p)′ → T (q),

such that h_{γ}(r) = F (r) ∩ T (q), which is a homeomorphism onto its image. Let d_{T (·)} denote the induced Riemannian path-metric on the plaque T (·).

Definition: With notation as above, the foliation F is θ-Hölder continuous with respect to T if for every R > 0, there is a constant H(R) > 0 and an ϵ > 0 such that for every p, q ∈ M and for every leafwise path γ from p to q of length l(γ) ≤ R,

d_{T (q)}(h_{γ}(r), h_{γ}(r′)) ≤ H(R) · d_{T (p)}(r, r′)^{θ},

for all r, r′ ∈ B_{ϵ}(p) ∩ T (p). The function R ↦ H(R) for R ≥ 0 is called a θ-Hölder constant assignment of F (relative to T ).

Remark: Changing the Riemannian metric on M has the effect of changing the constant assignment H in the above definition; that is, the property of a foliation being θ-Hölder continuous with respect to a transverse family is independent of metric. In the following proposition, the actual values of the constants H_{0} and H(R) are not important; the content of the statement is that they can be chosen uniformly over perturbations of φ_{1}.

Proposition 3.1 Let S be a closed, negatively-curved surface and let φ_{1} : T_{1}S → T_{1}S be the time-one map of the geodesic flow φ_{t} on the unit tangent bundle T_{1}S of S. Then for every θ < 1 there exists an H_{0} > 0, a function H : R_{+} → R_{+}, and a neighborhood U of φ_{1} in Diff^{2}(T_{1}S), such that for every f ∈ U,

1. The T f-invariant splitting T M = E^{u} ⊕ E^{c} ⊕ E^{s} obtained in Corollary 1.3 is θ-Hölder continuous, with Hölder constant H_{0}.

2. The strong unstable foliation W^{u}(f) is θ-Hölder continuous with respect to W_{loc}^{cs}(f), with θ-Hölder constant assignment R ↦ H(R).

3. The strong stable foliation W^{s}(f) is θ-Hölder continuous with respect to W_{loc}^{cu}(f) with θ- Hölder constant assignment R ↦ H(R).

Remark: At first glance, it might appear that the Hölder-continuity of W^{u}(f) follows from the fact that the leaves of W^{u}(f) are tangent to a Hölder-continuous line-field E^{u}, just as a Lipschitz vector field generates a Lipschitz flow. In the Appendix, we show that condition 2 does not follow from condition 1. In other words, a Hölder-continuous vector field, even if it is uniquely integrable, does not necessarily integrate to a Höldercontinuous flow.

This section is organized as follows. In Section 3.2, we prove a general theorem about the Hölder-continuity of an invariant section of a fiber-bundle map. Theorem 3.3 may be used wherever the Hölder Section Theorem in ([18]: Theorem 5.18(c)) can be applied. In this section, we use Theorem 3.3 to prove Proposition 3.1. Because the proof of Proposition 3.1 is lengthy, we postpone the verification of one of the hypotheses of Theorem 3.3 to the later Section 3.3. Proof of Proposition 3.1: Let M := T_{1}S and let T M = H^{u} ⊕H^{c} ⊕H^{s} be the Anosov splitting for φ_{t}. The arguments below are for unstable bundles and foliations; the same results follow for stable bundles and foliations by considering the inverse φ_{−1}.

Let η be the (unit) vector field generating φ_{t} and choose vector fields ξ_{1}, ξ_{2} spanning H^{u} and H^{s}, respectively, so that {η, ξ_{1}, ξ_{2}} is a unit volume frame for M. The bundles H^{u} and H^{s} are C^{1} by [8]. The original metric is C^{∞} and so consequently, the vector fields ξ_{1} and ξ_{2} can be chosen to be C^{1}. Since φ_{t} preserves the volume form and T_{p}φ_{t}(η(p)) = η(φ_{t}(p)), it follows that there is a function λ : M → (1, ∞) such that

T_{p}φ_{1}(ξ_{1}(p)) = λ(p) · ξ_{1}(φ_{1}(p)),

and

T_{p}φ_{1}(ξ_{2}(p)) = λ(p)^{−1} · ξ_{2}(φ_{1}(p)),

for all p ∈ M. These vector fields define a new C^{1} Riemannian structure on M in which {η, ξ_{1}, ξ_{2}} form an orthonormal frame. With respect to this metric,

∥T_{p}φ_{1}|_{H^{u}}∥ = λ(p), ∥T_{p}φ_{1}|_{H^{s}}∥ = λ(p)^{−1}, ∥T_{p}φ_{1}|_{H^{c}}∥ = 1.

All calculations that follow are done with respect to the new Riemannian structure on M.

These vector fields uniquely trivialize the bundles H^{u}, H^{c},H^{s}, H^{cs} and H^{cu}. Let α : M × R → H^{u} be a C^{∞} trivializing map for H^{u} such that for p ∈ M, α_{p} := α(p, ·) : R → H^{u}(p) is the linear isometry with α_{p}(1) = ξ_{1}(p). Similarly, let β be the trivializing map for H^{cs} such that β_{p} := β(p, ·) : R^{2} → H^{cs}(p) is orthonormal, with β_{p}((1, 0)) = ξ_{2}(p) and β_{p}((0, 1)) = η(p).

Now suppose f ∈ Diff^{2}(M). With respect to the splitting T M = H^{u} ⊕ H^{cs}, write:

T_{p}f = ( \begin{matrix}A_{p} & B_{p} \\ C_{p} & K_{p} \end{matrix} ) ,

where A_{p}, B_{p}, C_{p}, K_{p} are linear maps. If the approximation of f to φ_{1} is close in the C^{1} sense, then

∥B_{p}∥, ∥C_{p}∥ ≐ 0,
m(A_{p}) = ∥A_{p}∥ ≐ ∥T_{p}φ_{1}|_{H^{u}}∥, ∥K_{p}∥ ≐ ∥T_{p}φ_{1}|_{H^{c}}∥
and m(K_{p}) ≐ ∥T_{p}φ_{1}|_{H^{s}}∥,

uniformly in p ∈ M. (Note that A_{p} is a 1 × 1 matrix, so its norm and conorm are automatically equal).

The proof of 1. follows from standard graph-transform techniques and the Pointwise Hölder Section Theorem (Theorem 3.3). We prove Hölder-continuity of the splitting E^{u} ⊕ E^{c} ⊕ E^{s} in two steps. First we first show directly that the bundles E^{u} and E^{s} are Hölder-continuous. Then, to show that E^{c} is Hölder-continuous, we show that the weak stable and unstable bundles E^{cu} and E^{cs} are Hölder-continuous and then intersect them to obtain E^{c}.

Consider the bundle L → M with fiber L(H^{u}(p), H^{cs}(p)) over p, where L(H^{u}(p), H^{cs}(p)) is the space of linear maps from H^{u}(p) to H^{cs}(p), with the operator norm. Note that this bundle is trivial, since H^{u} and H^{cs} are trivial; the trivialization is given by

κ : M × L(R, R^{2}) → L
κ(p, P) = β_{p} ∘ P ∘ α_{p}^{−1}.

In fact L is metrically trivial: κ(p, ·) is an isometry with respect to the operator norms on L(R, R^{2}) and L(H^{u}(p), H^{cs}(p)). Let L(1) be the compact subbundle of L whose fibers are the linear maps of operator norm ≤ 1. κ induces a trivialization of L(1), as well.

We then define a bundle map Γ_{T f} : L(1) → L(1) covering f, which on each fiber {P ∈ L(H^{u}(p), H^{cs}(p)) | ∥P ∥ ≤ 1} is the linear graph transform:

Γ_{T f}(p, P) := (f(p), (C_{p} + K_{p} ∘ P) ∘ (A_{p} + B_{p} ∘ P)^{−1}).

If f is sufficiently C^{1}-close to φ_{1}, then Γ_{T f} will satisfy the hypotheses of Theorem 3.3. That is, in the notation of Section 3.2, there is an L > 0 such that if f ≐ φ_{1}, and δ > 0 is sufficiently small, then Γ_{T f} is Lipschitz with Lip(Γ_{T f}) ≤ L, and Γ_{T f} has pointwise fiber constant k_{p} (depending on f) and base constant μ_{p} (depending on f and δ) given by:

k_{p} ≐ \frac{∥K_{p}∥}{∥A_{p}∥} ≐ \frac{1}{∥T_{p}φ_{1}|_{H^{u}}∥} (since f ≐ φ_{1})

and

μ_{p} = Lip(f^{−1}|_{f(B_{δ}(p))})^{−1}
≐ m(T_{p}f)  (since δ ≐ 0)
≐ ∥T_{p}φ_{1}|_{H^{s}}∥ (since f ≐ φ_{1})

(where B_{δ}(p) denotes the ball in M centered at p of radius δ). But for each θ < 1, ∥T_{p}φ_{1}|_{H^{u}}∥ · ∥T_{p}φ_{1}|_{H^{s}}∥^{θ} > 1. Thus, if we initially fix δ > 0 small enough, then there is a neighborhood U_{0} of φ_{1} in Diff^{1}(M), such that for f ∈ U_{0},

k_{p} · μ_{p}^{−θ} < 1.

Then by Theorem 3.3 there is is an H_{0} > 0 such that, for f ∈ U_{0}, the unique Γ_{T f}- invariant section s : M → L of the bundle L is θ-Hölder continuous, with constant H_{0}.

This invariant section gives the unstable bundle E^{u} for f by: E^{u}(p) = {(x, s(p, x)) | x ∈ H^{u}(p)}, and so E^{u} is uniformly θ-Hölder continuous.

Similarly, we can find E^{cu} as the graph of an invariant section of a bundle over M, this time with fiber L(H^{cu}(p), H^{s}(p)). In this case we also have k_{p} ≐ ∥T_{p}φ_{1}|_{H^{u}}∥^{−1} and μ_{p} ≐ ∥T_{p}φ_{1}|_{H^{s}}∥, so it follows that for f sufficiently C^{1}-close to φ_{1}, the bundle E^{cu} is θ-Hölder continuous, with constant H_{0}, and the same is true for E^{cs}. Since the transverse intersection of θ-Hölder bundles is θ-Hölder, it follows that E^{c} is θ-Hölder and the proof of 1. is complete.

To prove 2., we will construct a nonlinear version of the bundle L. For ν > 0, denote by B_{ν}^{k}(x) the closed ball of radius ν, with respect to the box-norm, centered at x in R^{k}. Define the map

ω : M × R^{3} → M

by:

ω_{p}(t_{1}, t_{2}, t_{3}) := exp_{p}((α_{p}(t_{1}) + β_{p}(t_{2}, t_{3})),

where exp is the C^{∞} exponential map with respect to the C^{∞} canonical (Sasake) metric on M. Note that T_{0}ω_{p} is orthonormal with respect to the standard metric on R^{3} and the adapted C^{1} metric on M defined above. For ν_{0} sufficiently small, ω is a (C^{1}) diffeomorphism from B_{ν_{0}}^{3}(0) onto its image in M. In fact more is true: fixing a value of p, the functions ω_{p}(x) and ∂ω_{p}(x)/∂p are C^{∞} in x. A function with this property is said to be of class C^{1×∞}.

There is a well-defined bundle map

F : M × B_{ν_{0}}^{3}(0) → M × R^{3},

covering f : M → M. It is given by:

F (p, x) = (f(p), F_{p}(x)),

where

F_{p}(x) := ω_{f(p)}^{−1} ∘ f ∘ ω_{p}(x),

for p ∈ M, x ∈ B_{ν_{0}}^{3}(0).

Then

D_{0}F_{p} = T_{f(p)}ω_{f(p)}^{−1} ∘ T_{p}f ∘ T_{0}ω_{p}.

Since T_{0}ω_{p} maps the standard basis of R^{3} to the orthonormal basis {ξ_{1}(p), ξ_{2}(p), η(p)} of T_{p}M, we identify D_{0}F_{p} with T_{p}f so that by a slight abuse of notation:

D_{0}F_{p} = ( \begin{matrix}A_{p} & B_{p} \\ C_{p} & K_{p} \end{matrix} ) ,

Then F_{p} takes the form:

F_{p}(x, y) = (A_{p}x + B_{p}y, C_{p}x + K_{p}y) + r_{p}(x, y),  (1)
for p ∈ M, x ∈ B_{ν_{0}}^{1}(0), y ∈ B_{ν_{0}}^{2}(0)

where the remainder r : M × B_{ν_{0}}^{3}(0) → M × R^{3} is a C^{1×∞} function. Since D_{0}r_{p} = 0, for all p ∈ M, we may choose ν_{0} > 0 small enough so that Lip(r(p, ·)) ≐ 0, uniformly in p ∈ M.

For ν > 0 and g : B_{ν}^{1}(0) → R^{2} a continuous function with g(0) = 0, define the special norm:

∥g∥_{∗} := sup_{x∈B_{ν}^{1}(0)} \frac{|g(x)|}{|x|}.

(Naturally, this and all similar suprema are taken over x ≠ 0). With the norm ∥ ∥_{∗}, the space

G_{∗ν} := {g : B_{ν}^{1}(0) → R^{2} | g(0) = 0, ∥g∥_{∗} < ∞}

is a Banach space ([18], p. 62).

Now consider the subset G_{ν} ⊂ G_{∗ν} consisting of Lipschitz functions of Lipschitz norm ≤ 1:

G_{ν} := {g : B_{ν}^{1}(0) → R^{2} | g(0) = 0, Lip(g) ≤ 1}.

With respect to the metric given by ∥ ∥_{∗}, G_{ν} is a closed subset of the unit ball in G_{∗ν}. We will view the graph of an element of G_{∗ν} as a candidate for a local stable manifold. A choice of such a function g_{p} ∈ G_{ν} at each point p ∈ M is a section of the fiber bundle M × G_{ν}.

For ν_{1} < ν_{0} sufficiently small and or f sufficiently C^{1}-close to φ_{1}, the restriction of F_{p} to B_{ν_{1}}^{1}(0) satisfies the hypotheses of the pseudo-hyperbolic stable manifold theorem in [18]. (An outline of some the discussion that follows, on the level of the fibers G_{∗ν} is in [18], pages 61ff). Thus F induces a graph-transform bundle map F_{♯} : M×G_{ν_{1}} → M×G_{ν_{1}}, given by

F_{♯}(p, g) := (f(p), F_{♯}_{p}(g)),

where:

• For (p, g) ∈ M × G_{ν_{1}}, graph(F_{♯}_{p}(g)) = F_{p}(graph(g)) ∩ B_{ν_{1}}^{3}(0).

• Lip(F_{♯}_{p}(·)) ≐ ∥K_{p}∥ · m(A_{p})^{−1} (since ν_{1} ≐ 0) ≐ ∥T_{p}φ_{1}|_{H^{u}}∥^{−1} (since f ≐ φ_{1}).

Proposition 3.4 shows that there is an L > 0 and a neighborhood U_{1} of φ_{1} in Diff^{2}(M) such that for f ∈ U_{1}, F_{♯} is Lipschitz, with Lipschitz norm L; that is, for g ∈ G_{ν_{1}} and q, q′ ∈ M,

∥F_{♯}_{q}(g) − F_{♯}_{q′}(g)∥_{∗} ≤ L · d(q, q′).

We now show that F_{♯} satisfies the hypotheses of the Pointwise Hölder Section Theorem (Theorem 3.3). First, F_{♯} is a Lipschitz bundle map, covering a overflowing bilipschitz homeomorphism f. For p ∈ M, F_{♯} has Lipschitz fiber constant k_{p} < 1 (depending on f and ν_{1}) and base constant μ_{p} > 0 (depending on f and δ) defined as follows. As in the linear case, we have:

μ_{p} = Lip(f^{−1}|_{f(B_{δ}(p))})^{−1} ≐ m(T_{p}f) ≐ ∥T_{p}φ_{1}|_{H^{s}}∥,

and by Lemma III.6 in [18],

k_{p} = Lip(F_{♯}_{p}(·)) ≐ \frac{1}{∥T_{p}φ_{1}|_{H^{u}}∥} < 1.

It follows that if δ and ν_{1} are chosen initially to be sufficiently small, then there is a neighborhood of φ_{1} in Diff^{1}(M) where the relation

sup_{p∈M} k_{p} · μ_{p}^{−θ} < 1 (∗)

will hold for F_{♯}.

Given θ < 1, and δ, ν_{1} > 0 small enough, let U ⊂ U_{0} ∩ U_{1} be a neighborhood of φ_{1} in Diff^{2}(M) such that relation (∗) is satisfied, for every f ∈ U. By Theorem 3.3, for f ∈ U there is a unique continuous function s_{f} : M → G_{ν_{1}} such that:

F_{♯}(p, s_{f}(p)) = (f(p), s_{f}(f(p))).

Further, there is an H_{1} > 0 such that for all f ∈ U and for every p, q, ∈ M,

∥s_{f}(p) − s_{f}(q)∥_{∗} ≤ H_{1} · d(p, q)^{θ}.

Fix f ∈ U and write “s” for “s_{f}”. The graph of s(p) : B_{ν_{1}}^{1}(0) → R^{2} in B_{ν_{1}}^{3}(0) maps, under ω_{p}, to the local strong unstable manifold for f at p. For p ∈ M, let σ_{p} : B_{ν_{1}}^{1}(0) → M be given by:

σ_{p}(t) := ω_{p}(t, s(p)(t)).

Then σ_{p}(B_{ν_{1}}^{1}(0)) = W_{loc}^{u}(p).

We show now that the Hölder-continuity of s implies the Hölder-continuity of the foliation W^{u} with respect to plaque family W_{loc}^{cs}. For ν_{2} < ν_{1} sufficiently small, the Riemannian path - metric on ω_{p}(B_{ν_{2}}^{3}(0)) is uniformly close to the Euclidean metric on B_{ν_{2}}^{3}(0). In local ω_{p}-coordinates, let T_{p}(q) be the plane through q ∈ ω_{p}(B_{ν_{2}}^{3}(0)) given by:

T_{p}(q) := {ω_{p}(ω_{p}^{−1}(q), t_{1}, t_{2}) | (t_{1}, t_{2}) ∈ B_{ν_{2}}^{2}(0)}.

First consider the plaque family of transversals T = {T (p)}_{p∈M} given by:

T (p) := T_{p}(p) = {ω_{p}(0, t_{1}, t_{2}) | (t_{1}, t_{2}) ∈ B_{ν_{2}}^{2}(0)}.

The change of coordinates

Ω_{p,q} : B_{ν_{2}}(p) ∩ B_{ν_{2}}(q) → B_{ν_{2}}(q)

given by Ω_{p,q}(x) = ω_{q} ∘ ω_{p}^{−1}(x) is uniformly C^{1×1×∞} in the variables (p, q, x) and as d(p, q) → 0, Ω_{p,q} \overrightarrow{→} Id. It follows that as d(p, q) → 0,

T (p) \overrightarrow{→} T (q)

Figure 8: The W^{u}-holonomy between T (p) and T (p′).

and that the angle between T_{p}(q) and T (q) is bounded by c_{0} · d(p, q), where c_{0} is a uniform constant. In what follows we write T (p)(ϵ) := T (p) ∩ B_{ϵ}(p). Similarly define the “x-axis” for the ω_{p}-coordinate system by:

X(q) := {ω_{q}(t, 0, 0) | t ∈ B_{ν_{1}}^{1}(0)}.

Fix p for now and suppose p′ ∈ W_{ν_{2}}^{u}(p). Then p′ = σ_{p}(x), for some x ∈ B_{ν_{2}}^{1}(0). Consider the W^{u}-holonomy between the transversals T (p) and T (p′):

h^{u} : T (p)(ν_{2}) → T (p′).

T (p′) is nearly parallel to T_{p}(p′) (the angle between them is ≤ c_{0} · d(p, p′)) and so there exists a uniform constant c_{1} > 0 such that:

d_{T (p′)}(h^{u}(p), h^{u}(q)) = d_{T (p′)}(p′, h^{u}(q)) ≤ c_{1} · d_{T_{p}(p′)}(p′, q′),

where q′ = W_{ν_{2}}^{u}(q) ∩ T_{p}(p′) (See Figure 8).

The x-axes X(p) and X(q) are nearly (in a Lipschitz sense) parallel, so there exists a uniform constant c_{2} > 0 such that

d_{T_{p}(p′)}(p′, q′) ≤ c_{2} · d(σ_{p}(x), σ_{q}(x)).

Finally, since T (p) and T (q) are nearly parallel, there exists c_{3} > 0 such that

d(σ_{p}(x), σ_{q}(x)) ≤ c_{3} · [d_{T (p)}(p, q) + |s_{p}(x) − s_{q}(x)|]
≤ c_{3} · [d_{T (p)}(p, q) + |x| · ∥s_{p} − s_{q}∥_{∗}]
≤ c_{3} · [d_{T (p)}(p, q) + |x| · H_{1} · d(p, q)^{θ}],

Figure 9: d(p, q) ≤ c_{5} · d(p, q_{p}).

and so there exists a c_{4} > 0 such that:

d_{T (p′)}(h^{u}(p), h^{u}(q)) ≤ c_{4} · d_{T (p)}(p, q)^{θ}.

Now consider the center-stable plaque family W_{ν_{2}}^{cs}(f) = {W_{ν_{2}}^{cs}(p)}. This family is invariant under f and coherent: the plaques patch together to give a foliation of M. This property will be used in what follows. As f → φ_{1} in the C^{1} sense,

W_{ν_{2}}^{cs}(f) \overrightarrow{→} H_{ν_{2}}^{cs},

where H_{ν_{2}}^{cs} is there local center-stable plaque family for φ_{1}. The plaque H_{ν_{2}}^{cs}(p) is tangent to T (p). Locally, W^{cs}(p) is the graph of a smooth function:

g^{cs}(p) : T (p) → X(p).

(Where here we really mean that W^{cs}(p) = exp_{p}(graph(g^{cs}′(p))), where g^{cs}′(p) : H^{cs}(p) → H^{u}(p)). Further, there exists a small constant e < 1 such that if f ≐ φ_{1} in the C^{1} sense, then Lip(g^{cs}(p)) ≤ e, for all p ∈ M. For q ∈ T (p), let q_{p} = W_{ν_{2}}^{cs}(p) ∩ W_{ν_{2}}^{u}(q). Claim: Let c_{5} = (1 + e) · (1 − e)^{−1}. Then:

c_{5}^{−1} · d(p, q_{p}) ≤ d(p, q) ≤ c_{5} · d(p, q_{p}).

Proof of Claim: See Figures 9 and 10. Let q_{1} denote the intersection of X(q) with T (p). We use the facts that Lip(s(q)) ≤ 1 and Lip(g^{cs}′(p)) ≤ e.

Figure 10: d(p, q_{p}) ≤ c_{5} · d(p, q).

On the one hand,

d(p, q) ≤ d(p, q_{1}) + d(q_{1}, q)
≤ d(p, q_{1}) + e · d(p, q_{1})
≤ (1 + e) · d(p, q_{p})
≤ c_{5} · d(p, q_{p}),

since d(p, q_{1}) ≤ d(p, q_{p}) and d(q, q_{1}) < e · d(p, q_{1}). On the other hand, by inspecting Figure 10, we see that

d(p, q_{p}) ≤ (1 + 2(e + e^{2} + e^{3} + ⋯)) · d(p, q)
= (1 + \frac{2e}{1 − e}) · d(p, q)
= c_{5} · d(p, q),

and the claim is proved.

To complete the proof of Proposition 3.1, let p ∈ M, q ∈ W_{ν_{2}}^{cs}(p), and p′ ∈ W_{ν_{2}}^{u}(p) and q′ ∈ W_{ν_{2}}^{cs}(p′) ∩ W_{ν_{2}}^{u}(q). Then

d_{W^{cs}(p′)}(p′, q′) ≤ c_{5} · d_{T (p′)}(p′, q′_{p′})
≤ c_{5} · c_{4} · d_{T (p)}(p, q_{p})^{θ}
≤ c_{5} · c_{4} · c_{5}^{θ} · d_{W^{cs}(p)}(p, q)^{θ}.

Thus for c = c_{5} · c_{4} · c_{5}^{θ}, the W^{u}-holonomy between W_{loc}^{cs}-transversals of distance ≤ ν_{2} apart is θ-Hölder continuous, with constant c.

Fix R ≥ 0 and suppose p′ ∈ W^{u}(p) and γ : [0, 1] → M, is a leafwise path from p to p′ of length l(γ) ≤ R. Since W^{u}(f)-leaves contract uniformly exponentially under iterates of f^{−1}, there exists a uniform (over U) k ∈ Z_{+} such that d(f^{−k}(p), f^{−k}(p′)) < ν_{2}. There also exists an ϵ > 0 such that if d(p, q) < ϵ, then for all f ∈ U, d(f^{−k}(p), f^{−k}(q)) < ν_{2}. Further, there is a D > 0 such that:

D^{−1} · d(p, q) ≤ d(f^{−k}(p), f^{−k}(q)) ≤ D · d(p, q),

for all f ∈ U. Consider the holonomy

h_{γ} : W_{ϵ}^{cs}(p) → W^{cs}(p′).

From the above considerations plus the invariance of W^{cs}(f) under f, we have

d_{W^{cs}(p′)}(h_{γ}(q_{1}), h_{γ}(q_{2})) ≤ D · d_{W^{cs}(f^{−k}(p′))}(h_{f^{−k}∘γ}(f^{−k}q_{1}), h_{f^{−k}∘γ}(f^{−k}(q_{2})))
≤ D · c · d_{W^{cs}(f^{−k}(p))}(f^{−k}(q_{1}), f^{−k}(q_{2}))^{θ}
≤ D · D^{θ} · c · d_{W^{cs}(p)}(q_{1}, q_{2})^{θ}.

Setting H(R) = D · D^{θ} · c, the proposition is proved.□

3.2 The Pointwise Hölder Section Theorem

Let (X, d) be a complete metric space and let Y ⊂ E be a closed bounded subset of diameter D in the Banach space E. We shall be concerned with sections of the trivial bundle X × Y over X. A section of such a bundle can naturally be regarded as a map s : X → Y. Given constants 0 < δ, θ < 1, and H > 0, we define the section spaces:

Γ_{δ,θ,H} := {s : X → Y | d(x, x′) ≤ δ ⇒ |s(x) − s(x′)| ≤ Hd(x, x′)^{θ}}
Γ_{θ,H} := {s : X → Y | |s(x) − s(x′)| ≤ Hd(x, x′)^{θ}}.

Endow both spaces with the uniform topology. The second space Γ_{θ,H} is the space of all θ-Hölder-continuous maps from X to Y with Hölder constant ≤ H.

Lemma 3.2 Given θ, δ as above. If H > D · δ^{−θ} then Γ_{δ,θ,H} = Γ_{θ,H}.

Proof of Lemma 3.2: Evidently Γ_{θ,H} ⊂ Γ_{δ,θ,H}. Suppose that s ∈ Γ_{δ,θ,H}. We want to show that:

\frac{|s(x) − s(x′)|}{d(x, x′)^{θ}} ≤ H.

This is true for d(x, x′) ≤ δ because s belongs to Γ_{δ,θ,H}. For d(x, x′) ≥ δ,

\frac{|s(x) − s(x′)|}{d(x, x′)^{θ}} ≤ \frac{diam(Y)}{δ^{θ}} = \frac{D}{δ^{θ}} < H.

□

Now suppose that F : X × Y → X × Y is a homeomorphism that preserves the set of fibers {{x} × Y }. Then F takes the form:

F (x, y) = (h(x), v(x, y)).

We assume the following hypotheses on F:

1. The map h : X → X is an overflowing bilipschitz homeomorphism, and there exists a δ > 0 such that for all x ∈ X

inf_{x′≠x}\{\frac{(h(x), h(x′))}{d(x, x′)} s.t. d(x, x′) < δ\} = μ_{x} > 0,

and μ := inf μ_{x} > 0.

2. There exists a constant L ≥ 1 such that for all x, x′ ∈ X and all y ∈ Y,

|v(x, y) − v(x′, y)| ≤ Ld(x, x′)^{θ}.

3. For each x ∈ X,

inf \frac{|v(x, y) − v(x, y′)|}{|y − y′|} = k_{x} < 1,

and sup_{x∈X} k_{x} = k < 1.

The map F induces a map F̃ on sections of the bundle X × Y → X by:

F̃s(h(x)) = v(x, s(x)).

Then F̃ carries continuous sections into continuous sections and contracts fibers of X × Y by a definite amount k < 1. Since the space sections is complete there exists a unique invariant section s̃ : X → Y satisfying: s̃(x) = F̃s̃(x) = v(h^{−1}(x), s̃(h^{−1}(x))). This section is continuous. The following theorem gives sufficient conditions for the invariant section to be Hölder (compare with [18], Theorem 5.18).

Theorem 3.3 (Pointwise Hölder Section Theorem) Given F as above. Suppose that

sup_{x∈X} k_{x}μ_{x}^{−θ} = η < 1

Then the unique F̃-invariant section s̃ is θ-Hölder continuous. The θ-Hölder constant H may be chosen to be no smaller than

\frac{LD}{μδ^{θ}(1 − η)}.

Proof of Theorem 3.3: Let H = LD/(μδ^{θ}(1 − η)). We show that Γ_{θ,H} is mapped into itself by F̃. The uniqueness of s̃ in the space of continuous sections then implies the result.

Because H is not less than Dδ^{−θ}, we only need to show that Γ_{δ,θ,H} is carried into itself by F̃ (Lemma 3.2). So pick an arbitrary s ∈ Γ_{δ,θ,H}. Then F̃s(h(x)) = v(x, s(x)). We want to show that F̃s ∈ Γ_{δ,θ,H}; in other words, that

d(h(x), h(x′)) ≤ δ ⇒ |v(x, s(x)) − v(x′, s(x′))| ≤ Hd(h(x), h(x′))^{θ}.

So suppose that d(h(x), h(x′)) ≤ δ. Case 1: d(x, x′) ≥ δ. Then

\frac{|v(x, s(x)) − v(x′, s(x′))|}{d(h(x), h(x′))^{θ}} ≤ \frac{Diam(Y)}{(μd(x, x′))^{θ}}
≤ \frac{D}{(μδ)^{θ}} ≤ H.

Case 2: d(x, x′) < δ. Then

|v(x, s(x)) − v(x′, s(x′))| ≤ |v(x, s(x)) − v(x, s(x′))| + |v(x, s(x′)) − v(x′, s(x′))|
≤ k_{x}|s(x) − s(x′)| + Ld(x, x′)^{θ}
≤ k_{x}Hd(x, x′)^{θ} + Ld(x, x′)^{θ}

(Since d(x, x′) ≤ δ ⇒ |s(x) − s(x′)| ≤ Hd(x, x′)^{θ}).

= (k_{x}H + L)d(x, x′)^{θ}
≤ (k_{x}H + L)(\frac{1}{μ_{x}} d(h(x), h(x′)))^{θ}

(Since d(x, x′) ≤ δ ⇒ d(h(x), h(x′)) ≥ μ_{x}d(x, x′)).

≤ (H \frac{k_{x}}{μ_{x}^{θ}} + \frac{L}{μ_{x}})d(h(x), h(x′))^{θ}
≤ (Hη + \frac{L}{μ})d(h(x), h(x′)))^{θ}
≤ Hd(h(x), h(x′)))^{θ},

since H ≥ L/μ(1 − η).□

3.3 The Bundle Map F_{♯} is Lipschitz

Let F_{♯} be the bundle map defined in the proof of Proposition 3.1. By Lemma III.6 in [18], it is Lipschitz when restricted to fibers. In this section, we show that F_{♯} is globally Lipschitz. In particular, we have:

Proposition 3.4 (F_{♯} is uniformly Lipschitz on constant sections) There exists an L > 0 and a neighborhood U_{1} of φ_{1} in Diff^{2}(M) such that for all f ∈ U_{1},

∥F_{♯}_{q}(g) − F_{♯}_{q′}(g)∥_{∗} ≤ L · d(q, q′),

for all q, q′ ∈ M, and g ∈ G_{ν_{1}}.

Proposition 3.4 in turn follows from Lemma 3.5:

Lemma 3.5 There exists C > 0 and a neighborhood U_{1} of φ_{1} in Diff^{2}(M) such that for all f ∈ U_{1},

|F_{q}(x) − F_{q′}(x)| ≤ C · d(q, q′) · |x|,

for all q, q′ ∈ M, and x ∈ B_{ν_{1}}^{3}(0).

Proof of Lemma 3.5: Recall that F_{p} = ω_{f(p)}^{−1} ∘ f ∘ ω_{p}. The map ω is C^{1×∞}, and f is C^{2}, so F_{p}(x) is C^{1×∞} in (p, x). In particular, if f is sufficiently close to φ_{1} in the C^{2} sense, for a fixed p ∈ M, the Lipschitz norm of \frac{∂F_{p}}{∂p} (x) as a function of x is bounded by a uniform constant C_{1}. Note also that for every p ∈ M, F_{p} fixes the origin of R^{3}, so in local coordinates,

\frac{∂F_{p}}{∂p} (0) = 0.

Since \frac{∂F_{p}}{∂p} (x) is C^{∞} as a function of x, with Lipschitz norm bounded by C_{1}, we have for x ∈ B_{ν_{1}}^{3}(0),

\frac{∂F_{p}}{∂p} (x)|_{p=q} ≤ C_{1}|x|,

for all q ∈ M. But F is C^{1} as a function of p, so by the Mean Value Theorem, in local linear coordinates we have:

|F (q, x) − F (q′, x)| = | ∫_{0}^{1} \frac{∂F_{p}}{∂p} (x)|_{p=tq+(1−t)q′}dt| · |q − q′| ≤ C_{1} · |x| · |q − q′|,

and the result follows. □

Proof of Proposition 3.4: Let U_{1} be the neighborhood specified in Lemma 3.5 and let f ∈ U_{1}. For x_{1} ∈ B_{ν_{1}}^{1}(0), x_{2} ∈ B_{ν_{1}}^{2}(0), and q ∈ M, write F_{q}(x_{1}, x_{2}) = (F_{1,q}(x_{1}, x_{2}), F_{2,q}(x_{1}, x_{2})). Then, by definition of the graph-transform, for g ∈ G_{ν_{1}} we have ,

F_{♯}_{q}(g)(F_{1,q}(x, g(x))) = F_{2,q}(x, g(x)),

or in other words,

F_{♯}_{q}(g)(x) = (F_{2,q} ∘ G) ∘ (F_{1,q} ∘ G)^{−1}(x),

where G := (id, g) : B_{ν_{1}}^{1}(0) → R^{3}. Now, for q, q′ ∈ M, g ∈ G_{ν_{1}} and x ∈ B_{ν_{1}}^{1}(0), we have:

|F_{♯}_{q}(g)(x) − F_{♯}_{q′}(g)(x)| = |F_{2,q} ∘ G ∘ (F_{1,q} ∘ G)^{−1}(x) − F_{2,q′} ∘ G ∘ (F_{1,q′} ∘ G)^{−1}(x)|
≤ |F_{2,q} ∘ G ∘ (F_{1,q} ∘ G)^{−1}(x) − F_{2,q} ∘ G ∘ (F_{1,q′} ∘ G)^{−1}(x)| +
|F_{2,q} ∘ G ∘ (F_{1,q′} ∘ G)^{−1}(x) − F_{2,q′} ∘ G ∘ (F_{1,q′} ∘ G)^{−1}(x)|
≤ Lip(F_{2,q} ∘ G)|(F_{1,q} ∘ G)^{−1}(x) − (F_{1,q′} ∘ G)^{−1}(x)| +
C · d(q, q′)|G ∘ (F_{1,q′} ∘ G)^{−1}(x)|
≤ C_{2} · |(F_{1,q} ∘ G)^{−1} ∘ (F_{1,q′} ∘ G) ∘ (F_{1,q′} ∘ G)^{−1}(x) −
(F_{1,q} ∘ G)^{−1} ∘ (F_{1,q} ∘ G) ∘ (F_{1,q′} ∘ G)^{−1}(x)| + C_{3} · d(q, q′)|x|
≤ C_{2} · Lip((F_{1,q} ∘ G)^{−1})|(F_{1,q′} ∘ G) ∘ (F_{1,q′} ∘ G)^{−1}(x) −
(F_{1,q} ∘ G) ∘ (F_{1,q′} ∘ G)^{−1}(x)| + C_{3} · d(q, q′)|x|
≤ C_{4} · C · d(q, q′)|(F_{1,q′} ∘ G)^{−1}(x)| + C_{3} · d(q, q′)|x|
≤ C_{5} · d(q, q′)|x| + C_{3} · d(q, q′)|x|
= L · d(q, q′)|x|,

and so

∥F_{♯}_{q}(g) − F_{♯}_{q′}(g)∥_{∗} = sup_{x∈B_{γ}^{1}(0)} \frac{|F_{♯}_{q}(g)(x) − F_{♯}_{q′}(g)(x)|}{|x|}
≤ L · d(q, q′).

□

4 Stable Ergodicity

In this section, we develop the remaining tools necessary to prove the Main Theorem and then carry out the proof. We restate it here for convenience:

Theorem 4.1 (Main Theorem) If S is a closed, connected negatively-curved Riemannian surface, and if φ_{t} : T_{1}S → T_{1}S is the geodesic flow, then the time-one map φ_{1} is stably ergodic.

Here is an outline of the proof. The strategy from this point on is similar in spirit to that in [5], but the estimates there need refinement to work in a setting where the local structure varies from point to point. Let M = T_{1}S and suppose that f ∈ Diff_{ω}^{2}(M). If f is sufficiently C^{1} close to φ_{1}, then Corollary 1.3 and 1.4 apply, and there is a T f-invariant splitting:

T M = E^{u} ⊕ E^{c} ⊕ E^{s}

and invariant foliations W^{u}, W^{c}, W^{s}, W^{cu}, and W^{cs} tangent to E^{u}, E^{c}, E^{s}, E^{u} ⊕ E^{c}, and E^{s}⊕E^{c}. The foliations W^{u} and W^{s} are dynamically-defined; that is, for q ∈ W^{s}(p),

d(f^{k}(p), f^{k}(q)) → 0,

as k → ∞, and similarly for W^{u}. Consequently, if g : M → R is a continuous measurement on M, then

lim_{k→∞} |g(f^{k}(p)) − g(f^{k}(q))| = 0, ∀q ∈ W^{s}(p),

and

lim_{k→∞} |g(f^{−k}(p)) − g(f^{−k}(q′))| = 0, ∀q′ ∈ W^{u}(p).

Also, the foliations W^{u} and W^{s} are absolutely continuous, by [16]. If a foliation W is absolutely continuous, it makes sense to speak, at least locally, of “almost every leaf of W.” A measurable set A is essentially W-saturated if it essentially consists of essentially whole W-leaves.

If f is not ergodic, then a standard argument, originally due to Hopf (cf. [5], p. 297), shows that there is a measurable set A, with m(A) > 0 and m(M \ A) > 0, such that A is essentially W^{u}-saturated and essentially W^{s}-saturated. In what follows, we show how this gives a contradiction.

For such an f, let a be a density point of A, and let b be a density point of M \ A. In appropriately-chosen rectilinear coordinates, a small cube of side-length w centered at a intersects A in a set of very high density in the cube. Similarly for a small cube around b. The next step is the crucial one in [5]. Divide each cube into long, thin prisms of cross-sectional width w and height w^{2/3}. At least one of these prisms, say P, will have a high concentration of A. Similarly, there is prism P′ near b with a high concentration of M \ A. In Section 4, we show that, for f sufficiently C^{2}-close to φ_{1}, nested inside the prisms P and P′ are dynamical-prisms called juliennes J and J′, in which A and M \ A are also highly concentrated.

Now by Lemma 2.1, if f is sufficiently close to φ_{1} in the C^{1} sense, then there is a W^{u,s}-path τ consisting of three arcs of bounded length that joins the centers of J and J′. Again, if f is C^{2}-close to φ_{1}, then the W^{u} and W^{s} holonomy maps are uniformly α-Hölder-continuous, with α close to 1, and it follows that the holonomy maps don’t distort juliennes too much. In Subsection 4.6, we show that if an essentially W^{u}- (W^{s} -) saturated set is highly concentrated in J then its image under the W^{u}- (W^{s} -) holonomy is highly concentrated in another julienne. Applying this argument to A along successive legs of the path τ, we arrive at the conclusion A is highly concentrated in J′, a contradiction.

In Section 4.8 we prove Theorem 4.1.

4.1 Holonomy Revisited

Let V be the neighborhood of φ_{1} given in Lemma 2.1 and for f ∈ V, let N be a generous bound on the lengths of the three-legged W^{u,s}(f) paths given by that lemma. For p ∈ M and R ≤ N let γ : [0, 1] → W^{u}(p) be a leafwise geodesic with γ(0) = p of length l(γ) = R. Let h_{R}^{u} denote the holonomy

h_{R}^{u} := h_{γ}^{u} : H_{loc}^{cs}(p) → H_{loc}^{cs}(γ(1)).

Similarly, for f close to φ_{1} in the C^{2} sense, let π_{R}^{u} denote the W^{u}(f)- holonomy between W_{loc}^{cs}(f)-transversals of leafwise distance R apart. (Incidentally, the discussion that follows holds equally for the W^{s}-holonomy, π_{R}^{s}). In Chapter 3 we showed that for R bounded, π_{R}^{u}, while not necessarily smooth, is uniformly α-Hölder with α near 1. In this section, we investigate further the properties of π_{R}^{u}.

4.2 Absolute Continuity

Pre-perturbation, the unstable holonomy h_{R}^{u} is smooth because the foliation H^{u} is C^{1}. This implies that the Jacobian of h_{R}^{u} is uniformly bounded above and below. In this subsection we show that for f ≐ φ_{1} in the C^{2} sense, and for R bounded, the Jacobian Jac(π_{R}^{u}) exists, is continuous and uniformly bounded above and below (away from 0).

Lemma 4.2 (Unstable holonomy Jacobians converge uniformly) As f → φ_{1} in the C^{2} sense, π_{R}^{u} converges uniformly to h_{R}^{u} and Jac(π_{R}^{u}) converges uniformly to Jac(h_{R}^{u}).

Proof of Lemma 4.2: As f → φ_{1} in the C^{2}-sense, the local disk families W_{loc}^{cs}(f) converge uniformly to H_{loc}^{cs} (see the proof of Proposition 3.1), hence we have

π_{R}^{u} \overrightarrow{→} h_{R}^{u}.

To show Jac(π_{R}^{u}) \overrightarrow{→} Jac(h_{R}^{u}), we first show that Jac(π_{R}^{u}) is uniformly bounded in a neighborhood of φ_{1} in Diff^{2}(M). The convergence then follows from the Lebesgue dominated convergence theorem (compare [5], Lemma 2.2).

The continuity of Jac(π_{R}^{u}) is proved in [16]. Inspecting the proof of Theorem 2.1 in [16], we see that the value of Jac_{p}(π_{R}^{u}) is given by the limit:

unif lim \frac{det(f^{−n}|_{T_{y}D_{p}})}{det(f^{−n}|_{T_{y}D_{q}})},

which in turn is bounded by:

unif lim \frac{det(T f^{−n}|_{T_{y}D_{p}})}{det(T f^{−n}|_{T_{y}D_{q}})},

which, by the chain rule, is bounded by a function of:

∑_{k=0}^{∞} | det(T_{f^{−k}(p)}^{cs}f^{−1}) − det(T_{f^{−k}(q)}^{cs}f^{−1})|.

Finally, this expression is bounded by

∑_{k=0}^{∞}(λ^{−θ})^{k} · d(p, q),

where λ = inf_{x} m(T_{x}f|_{E^{u}}) and θ is the Hölder exponent of the distribution E^{u}. Since these quantities vary uniformly in a neighborhood U of φ_{1} in Diff^{1}(M), it follows that the Jacobian Jac(π_{R}^{u}) is uniformly bounded in U.□

When S has constant curvature, the horocyclic flow preserves the volume form, and consequently Jac(h_{R}^{u}) is identically equal to 1, regardless of the distance R between transversals. Due to metric rigidity, this is probably never the case when the curvature of S is variable. Nonetheless, its Jacobian is continuous, and thus uniformly bounded if R is bounded; Lemma 4.2 then implies that the value of Jac(π_{R}^{u}) is uniformly bounded, and we have:

Lemma 4.3 (Uniform bounds for unstable holonomy Jacobians) Given N, there is a neighborhood N_{1} of φ_{1} in Diff^{2}(M) and a constant j > 1 such that for all f ∈ N_{1}, if R ≤ N, then

j^{−1} < Jac(π_{R}^{u}) ≤ j.

4.3 Behavior Along W^{c} Leaves

Restricted to flow lines H^{c}, the holonomy h_{R}^{u} is an isometry. This is just a restatement of the fact that the Anosov splitting is invariant under T φ_{t} and the flow φ_{t} is an isometry along center leaves. Post-perturbation, nearly the same is true. We have:

Lemma 4.4 (cf. [5], Lemma 2.3) The W^{u}(f)-holonomy map π_{R}^{u} preserves the center foliation W^{c}. If p′ = π_{R}^{u}(p), then π_{R}^{u} sends W_{loc}^{c}(p) to W_{loc}^{c}(p′), is C^{1} and is uniformly nearly isometric when restricted to W_{loc}^{c}(p).

Proof of Lemma 4.4: The proof carries through exactly as in [5].□

4.4 Juliennes and Rectangles

In this section we recall some of the material in [5], sections 4-5. Much of the material in these sections carries over mutatis mutandis to our setting. Here f is a C^{2} volumepreserving perturbation of the time-one map φ_{1} of the geodesic flow on M = T_{1}S, where S is a closed, negatively-curved surface.

4.5 Definitions

Let T M = E^{u} ⊕ E^{c} ⊕ E^{s} be the T f-invariant splitting of T M into unstable, center and stable directions. This splitting is Hölder continuous. For each p ∈ M, choose smooth (C^{∞}) local coordinates (x, y, z)_{p} on M so that

p = (0, 0, 0)_{p}, span (\frac{∂}{∂x})_{p} = E_{p}^{u}, span (\frac{∂}{∂y})_{p} = E_{p}^{s}, span (\frac{∂}{∂z})_{p} = E_{p}^{c}.

Let Z_{p} be the plane z = 0.

A rectangle centered at p_{0} = (x_{0}, y_{0}, 0)_{p} of width 2w and height 2h is a rectilinear prism in p-coordinates:

R(p, p_{0}, w, h) = [x_{0} − w, x_{0} + w] × [y_{0} − w, y_{0} + w] × [−h, h].

A square S(p, p_{0}, w) = [x_{0} − w, x_{0} + w] × [y_{0} − w, y_{0} + w] is the Z_{p}- slice of a rectangle. Rectangles have minimal dynamical significance, but they can be used to compose rectilinear cubes and hence are useful in the context of the Lebesgue Density Theorem.

By contrast, juliennes are dynamically-constructed approximations to rectangles. As we shall see, juliennes behave well under holonomy maps. They are constructed as follows. We first alter the coordinate axes in the Z_{p} plane. The laminations W^{cu}, W^{cs} give transverse laminations of Z_{p}:

L^{u} = W^{cu} ∩ Z_{p}, L^{s} = W^{cs} ∩ Z_{p}.

The leaves L^{u}, L^{s} of these laminations form Hölder coordinates for Z_{p}.

Let L^{u}(p_{0}) and L^{s}(p_{0}) be, respectively, the intersections of L_{loc}^{u}(p_{0}) and L_{loc}^{s}(p_{0}) with the square S(p, p_{0}, w). For w small enough, L^{u}(p_{0}) and L^{s}(p_{0}) are smooth curves transverse to each other which define coordinate axes for the square

Σ(p, p_{0}, w) = ⋃ L_{loc}^{u}(r) ∩ L_{loc}^{u}(s),

where r and s range over L^{u}(p_{0}) and L^{s}(p_{0}) respectively.

The nonlinear square Σ(p, p_{0}, w) forms the base of a julienne

J(p, p_{0}, w, h) = ∪_{r∈Σ(p,p_{0},w)}W_{h}^{c}(r),

where W_{h}^{c}(r) is the arc of W_{loc}^{c}(r) lying between the planes z = ±h.

The center foliation arcs comprising the vertical direction of a julienne are integral curves of a Hölder-continuous vector field and so we have some control over how vertical strips deviate horizontally. The square Σ(p, p_{0}, w), on the other hand, is nearly tangent to the very non-integrable distribution E^{u} ⊕ E^{s} (pre-perturbation, this distribution is totally non-integrable), so we can’t hope to control the vertical deviation of a horizontal strip if w is too large relative to h. It turns out that if w is sufficiently small relative to h, then juliennes will approximate rectangles.

More precisely, under appropriate conditions, a julienne will contain a scaled copy of a rectangle and vice versa. By a scaled julienne we mean: fix a real number c ∈ (0, 1) and let

cR = R(p, p_{0}, cw, ch), and cJ = J(p, p_{0}, cw, ch).

The next lemma shows that if w and h are related by: w = k · h^{3/2}, where k > 0 is a constant bounded above and below from zero, then the associated families of rectangles and juliennes are comparable at small w-scales. Assume that p_{0} ∈ M lies in the coordinate systems at both p_{1} and p_{2}. There are four different objects to compare:

R_{1} = R(p_{1}, p_{0}, w, h) J_{1} = J(p_{1}, p_{0}, w, h)
R_{2} = R(p_{2}, p_{0}, w, h) J_{2} = J(p_{2}, p_{0}, w, h).

Lemma 4.5 (Julienne Nesting Lemma) (cf. [5], Lemma 3.1) Fix constants c ∈ (0, 1) and k_{2} > k_{1} > 0. As f → φ_{1} in the C^{2} sense and w = kh^{3/2} → 0 with k ∈ [k_{1}, k_{2}] and while h^{−1} · d(p_{0}, p_{1}) and h^{−1} · d(p_{0}, p_{2}) stay bounded, then:

cR_{1} ⊂ J_{2} ⊂ c^{−1}R_{1} and cR_{2} ⊂ J_{1} ⊂ c^{−1}R_{2}.

Proof of Lemma 4.5: We can choose a neighborhood N_{0} of f in Diff^{2}M such that the center line field E^{c} for f is uniformly (in the sense of Proposition 3.1) α-Hölder continuous with α > \frac{1}{2}. Since the adapted local coordinates at p are C^{∞}, the proof carries through as in [5].□

4.6 Density Estimates

Let A be an essentially W^{u}-saturated and denote by A^{u} the unstable saturate of A. It consists entirely of whole W^{u}-leaves – those that meet A in a set of full leaf measure. Then A and A^{u} differ by a zero-set, so m(A ∩ R) = m(A^{u} ∩ R) for any measurable set R. Define the conditional measure of a set A relative to R (with 0 < m(R) < ∞) in the standard way:

m(A : R) = \frac{m(A ∩ R)}{m(R)}.

Because we will be swapping between rectangles and juliennes, we need an estimate of how density decays in the process. Lemma 4.5 allows us to swap the two at small scales with little sacrifice in measure. Since we will slide juliennes along W^{u} leaves, preserving the W^{cs} lamination, it is convenient to concentrate on the density of a set A inside a W^{cs}-leaf. The strategy in [5] is to study first the density of A in a rectilinear approximation of the W^{cs}-leaf, and then, using Lemma 4.5, to make a similar statement about actual W^{cs}-leaves. In this way, the non-absolute continuity of the W^{cs} can be compensated for by approximating by a smooth foliation at small scales.

The non-linear center stable slice of a julienne J = J(p, p_{0}, w, h) is defined:

J_{x}^{cs} = W_{loc}^{cs}(x) ∩ J,

for x in the L^{u}-leaf L_{w}^{u}(p_{0}). The center stable midslice is the the leaf J^{cs} = J_{p_{0}}^{cs}.

Lemma 4.6 (cf. [5], Lemma 3.3) Fix ρ′ ≤ 1 and k_{2} > k_{1} > 0. Let d ∈ (0, 1) and .1 ≤ ρ ≤ ρ′ be given. If k ∈ [k_{1}, k_{2}], and w = kh^{3/2} is small enough and if f C^{2}- approximates φ_{1} well enough then for any essentially W^{u}-saturated set A,

m(A : J) ≥ ρ ⇒ m(A^{u} : J^{cs}) ≥ dρ,
m(A^{u} : J^{cs}) ≥ ρ ⇒ m(A : J) ≥ dρ.

Proof of Lemma 4.6: See [5]. There is a slight error in the proof there: they say that the mutual densities of R^{cs} and J^{cs} are nearly 1. This is clearly not true, but the Lemma follows easily from the fact that Jac(π_{R}^{u}) \overrightarrow{→} 1 as R → 0.□

The next lemma is essential for proving the estimates in Section 4.1.

Lemma 4.7 (cf. [5], Lemma 3.4) Fix k_{2} > k_{1} > 0. As w = kh^{3/2} → 0 with k ∈ [k_{1}, k_{2}] and f → φ_{1} in the C^{2} sense,

\frac{m(J^{cs})}{4wh} \overrightarrow{→} 1,

where J = J(p, p, w, h) and m denotes the area on the leaf W_{loc}^{cs}(p). Also, if J_{+}^{cs} denotes the part of J^{cs} lying to the right of W_{h}^{c}(p), then

\frac{m(J_{+}^{cs})}{2wh} \overrightarrow{→} 1.

Proof of Lemma 4.7: The proof is the same as in [5]; in our case, Jac(π^{u}) ≐ 1 because Jac(π_{R}^{u}) \overrightarrow{→} 1 as R → 0.□

4.7 Holonomy and Juliennes

Again let J^{cs}(p, q, w, h) denote the center-stable midslice of the julienne J(p, q, w, h).

Lemma 4.8 (Julienne Holonomy Lemma) Fix constants R ≥ 0, k_{2} > k_{1} > 0 and c ∈ (0, 1). Fix p ∈ M, and let j_{p} = Jac_{p}(h_{R}^{u}) and q = π_{R}^{u}(p). As f → φ_{1} in the C^{2} sense and w = kh^{3/2} → 0, with k ∈ [k_{1}, k_{2}],

cJ^{cs}(q, q, j_{p} · w, h) ⊂ π_{R}^{u}(J^{cs}(p, p, w, h)) ⊂ c^{−1}J^{cs}(q, q, j_{p} · w, h).

Proof of Lemma 4.8:(compare [5], Lemma 4.1) To simplify notation, let J denote the julienne J(p, p, w, h) and let J′ denote J(q, q, j_{p} · w, h). Similarly define J^{cs}, J^{′cs}, J_{+}^{cs}, and J_{+}^{′cs}. By Lemma 4.4, the image of J^{cs} under π_{R}^{u} is a union of center manifolds W_{loc}^{c}(q), each with length approximately equal to 1. This alone does not ensure that the image of J^{cs} does not look very ragged along the top. For example, with this information alone, the image of J^{cs} could resemble the image in Figure 11. Nearly half of the density of julienne in that figure has disintegrated.

Fortunately, we have the additional information from Proposition 3.1 that for f ≐ φ_{1} the holonomy π_{R}^{u} is uniformly α-Hölder continuous, with α > \frac{2}{3}, as is its inverse. This implies that the picture in Figure 11 cannot occur, for if the width of the top edge of J^{cs} is w, then the width of its image is ≤ H · w^{α}, and

Hw^{α} << (wk^{−1})^{\frac{2}{3}} = h

at small w-scales (since k is bounded). Set e = \sqrt{c}. For w small enough the image of the top boundary of J^{cs} lies between the planes z = eh and z = e^{−1}h in the (x, y, z)_{p′}- coordinate system. Similarly, the image of the bottom boundary of J^{cs} lies between the planes z = −eh and z = −e^{−1}h. See Figure 12. Now π_{R}^{u} is absolutely continuous by Lemma 4.2 and its Radon-Nikodym derivative Jac(π_{R}^{u}) is nearly constant and equal to j_{p}. Let b = \sqrt{e}. For w small enough, it follows that we have:

b · j_{p} ≤ Jac_{q}(π_{R}^{u}) ≤ b^{−1} · j_{p},

Figure 11: Center lines are preserved, but the top and bottom are ragged. Figure 12: The image of J^{cs} in the p′-coordinate system. for all q ∈ J^{cs}. Also, π_{R}^{u} is orientation-preserving, so the right half-julienne J_{+}^{cs} is mapped to the right half of W_{loc}^{cs}(p′).

By Lemma 4.4, π_{R}^{u} carries the rightmost edge of J_{+}^{cs} to another center manifold W_{loc}^{c}(r), that lies either wholly to the left or wholly to the right of the rightmost edge W_{loc}^{c}(q′) of J^{′cs}. Suppose that r lay to the left of the point cq′, so that π_{R}^{u}(J_{+}^{cs}) lay entirely in the positive region bounded by W_{loc}^{c}(cq′) and the planes z = ±e^{−1}h. By Lemma 4.7, the area of this region is well-approximated by :

c · (j_{p}w) · (2e^{−1}h) = 2ewhj_{p}.

But this region contains π^{u}(J_{+}^{cs}), which has area well-approximated by:

2wh · (j_{p} · b),

which gives a contradiction, since b > e.

Suppose instead that r lies to the right of the point c^{−1}q′, so that π_{R}^{u}(J^{cs}) wholly contained the the positive region bounded by W_{loc}^{c} and the planes z = ±eh. The area of this region is well-approximated by

c^{−1} · (j_{p}w) · (2eh) = 2e^{−1}whj_{p}.

But the area of π^{u}(J_{+}^{cs}) is well-approximated by:

2wh · (j_{p} · b^{−1}),

again giving a contradiction. The same argument applied to the left side of the julienne implies the result.□

Lemma 4.9 (Julienne Holonomy Density Lemma) Fix ρ′ < 1. Let c, ρ ∈ (0, 1) be constants such that .1 ≤ cρ < ρ ≤ ρ′ and let f, φ_{1}, R,k_{1},k_{2}, j_{p}, p, q, w and h have the same meanings as in Lemma 4.8. If k ∈ [k_{1}, k_{2}] and w = kh^{3/2}, is small enough, if f C^{2} approximates φ_{1} well enough, and if A is an essentially W^{u}-saturated set which has density ≥ ρ in J(p, p, w, h), then A has density ≥ cρ in J(q, q, j_{p} · w, h).

Proof of Lemma 4.9:(compare [5], Lemma 4.2) Let c, ρ > .1 be given and pick d so that 1 > d > c^{1/4} and 1 + 10(1 − d^{−1}) > \sqrt{c}. Note that since ρ ≥ .1, this implies that 1 − d^{−1}(1 − dρ) > \sqrt{c}ρ. Let J, J′ etc. be as in the proof of Lemma 4.8. Since A is essentially W^{u}-saturated, Lemma 4.6 implies that if f is close to φ_{1} in the C^{2} sense and w is small enough, then we will have:

m(A^{u} : J^{cs}) ≥ dρ,

or, in other words:

m(A^{u} ∩ J^{cs}) ≥ dρ · m(J^{cs}).

Denote by A^{u}′ the complement M \ A^{u} and note that A^{u}′ is the unstable saturate of M \ A. We have that:

m(A^{u}′ ∩ J^{cs}) ≤ (1 − dρ)m(J^{cs}).

Since A^{u}′ is W^{u}-saturated, π_{R}^{u} sends A^{u}′ ∩ J^{cs} to A^{u}′ ∩ π_{R}^{u}(J^{cs}). By Lemma 4.2, Jac(π_{R}^{u}) is nearly constant and equal to j_{p}, and so:

d^{1/2}j_{p} ≤ Jac_{q}(π_{R}^{u}) ≤ d^{−1/2}j_{p},

for all q ∈ J^{cs}. This in turn implies that:

m(A^{u}′ ∩ π_{R}^{u}(J^{cs})) ≤ d^{−1/2}j_{p}(1 − dρ)m(J^{cs}),

or, in other words:

m(A^{u}′ : π_{R}^{u}(J^{cs})) ≤ \frac{j_{p}(1 − dρ)m(J^{cs})}{\sqrt{d}m(π_{R}^{u}(J^{cs}))}.

Again, since Jac(π_{R}^{u}) is nearly constant, we have

\frac{j_{p} · m(J^{cs})}{m(π_{R}^{u}(J^{cs}))} ≤ d^{−1/2},

and so it follows that:

m(A^{u} : π_{R}^{u}(J^{cs})) ≥ 1 − d^{−1}(1 − dρ) ≥ \sqrt{c}ρ.

By Lemma 4.9, π_{R}^{u}(J^{cs}) and J^{′cs} are highly concentrated in each other, and so

m(A^{u} : J^{′cs}) ≥ d\sqrt{c}ρ.

Finally, Lemma 4.6 implies that:

m(A^{u} : J^{′cs}) ≥ d^{2}\sqrt{c}ρ = cρ.

Implicitly we are using here the fact that j_{p} is uniformly bounded above and below, so that these constants can be chosen uniformly at the same time for J and J′. The lemma is proved.□

The lemmas of this section hold equally for the π_{R}^{s}-holonomy maps.

4.8 Proof of Main Theorem

In this section we put together the results of the previous sections to prove the Main Theorem. The proof follows the lines of the proof of the main theorem in [5].

Suppose f is not ergodic. Then, as discussed in the beginning of this chapter, there is an essentially W^{u,s}(f)-saturated set A such that m(A) > 0 and m(M \ A) > 0. Let a be a density point of A and let b be a density point of M \ A.

Lemma 2.1 implies that if f is in a neighborhood V of φ_{1} in Diff^{2}(M), then there is a three-legged W^{u,s}-path of length ≤ N connecting any two points of T_{1}S. Lemma 4.3 implies that there exists a constant e ∈ (0, 1) such that for f ∈ V, if τ is any three-legged W^{u,s}(f)-path of length ≤ N, and if k represents the Jacobian of the concatenated π^{u}/π^{s} holonomy maps along τ, then

e ≤ k ≤ e^{−1}.

Pick constants ρ_{1}, d, ρ_{2} ∈ (0, 1), depending on e, as follows. First pick ρ_{1} > max{.1, 1 − e^{2}}. Now pick d < 1 so that

d > \frac{1 − e^{2}}{ρ_{1}} .

Now pick ρ_{2} < 1 such that

ρ_{2} > max{1 − e^{2}dρ_{1}, e^{−2}(1 − dρ_{1})}.

Choose h small enough so that the density of A in a rectilinear cube C_{a} about a of side-length 2h is ≥ \sqrt{ρ_{1}}. Also choose h so that the density of M \ A in a rectilinear cube C_{b} about b of side-length 2h is ≥ \sqrt{ρ_{2}}. Divide the cube C_{a} centered at a into rectilinear boxes of width 2w and height 2h, where w = h^{3/2}. Similarly, divide the cube C_{b} centered at b into rectangles of width 2w and height 2h. The density of A in one of these boxes, say R(a, a_{0}, w, h) is ≥ \sqrt{ρ_{1}}, and the density of M \ A in some box R(b, b_{0}, w, h) is ≥ \sqrt{ρ_{2}}. Lemma 4.5 then implies that A is highly concentrated in the julienne J_{a_{0}} := J(a_{0}, a_{0}, w, h) and M \ A is highly concentrated in J_{b_{0}} := J(b_{0}, b_{0}, w, h). We may assume, then, that we initially chose h small enough, so that m(A : J_{a_{0}}) ≥ ρ_{1} and m(A : J_{b_{0}}) ≥ ρ_{2}.

Let c = (d)^{\frac{1}{3}} , where d was chosen above, and assume we have chosen h small enough so that Lemma 4.9 holds for this value of c, with e < k < e^{−1}, ρ′ = \sqrt{ρ_{1}}, and with R = N. By Lemma 2.1, there is a three-legged W^{u,s}-path connecting a_{0} to b_{0} of length ≤ N. Let a_{0} = q_{0} → q_{1} → q_{2} → q_{3} = b_{0} be the W^{u,s}-path from a_{0} to b_{0} and let j_{q_{0}}, j_{q_{1}}, j_{q_{2}} be the Jacobians of the corresponding holonomy maps, with concatenated Jacobian k_{0} := j_{q_{0}}j_{q_{1}}j_{q_{2}}. Lemma 4.9 then implies that the density of A in J(q_{1}, q_{1}, j_{q_{0}}w, h) is at least cρ_{1}. Continuing along the W^{u,s}-path, applying Lemma 4.9 two more times, we arrive at the conclusion that the density of A in J(b_{0}, b_{0}, k_{0}w, h) is at least c^{3} · ρ_{1} = dρ_{1}. We show now that this gives a contradiction. There are two cases to consider. Suppose first that k_{0} < 1, so that the julienne J(b_{0}, b_{0}, k_{0}w, h) is contained in the julienne J(b_{0}, b_{0}, w, h) (see Figure 13). Lemma 4.5 then implies that the density of A in J(b_{0}, b_{0}, w, h) is well-approximated by (k_{0})^{2} · dρ_{1}, which in turn is well-approximated (below), by e^{2}dρ_{1}. Since the density of M \ A in J(b_{0}, b_{0}, w, h) is at least ρ_{2}, and ρ_{2} > 1 − e^{2}dρ_{1}, this gives a contradiction. Suppose on the other hand that k_{0} > 1, so that the julienne J(b_{0}, b_{0}, w, h) is contained in the julienne J(b_{0}, b_{0}, k_{0}w, h) (see Figure 14). The density of M \ A in J(b_{0}, b_{0}, k_{0}w, h) is bounded above by 1 − dρ_{1}, which, again by Lemma 4.5 implies that the density of M \ A in J(b_{0}, b_{0}, w, h) is less than e^{−2}(1 − dρ_{1}). This gives a contradiction, since ρ_{2} > e^{−2}(1 − dρ_{1}). The theorem is proved. □

Figure 13: k_{0} < 1: The density of A in J(b_{0}, b_{0}, w, h) is not too small.

Figure 14: k_{0} > 1: The density of M \ A in J(b_{0}, b_{0}, w, h) is not too large.

Figure 15: The graph y = f_{c}(x).

5 Appendix

In this section we construct a uniquely integrable Hölder continuous vector field whose integral foliation is not Hölder continuous.

Given real parameters c = (a, b), with 1 > b > a > 0, define f_{c} : R → R by:

f_{c}(x) = \{ \begin{matrix}a & if x < 0 \\ x^{2} + a & if 0 ≤ x < b \\ 2b^{2} + a − (x − 2b)^{2} & if b ≤ x < 2b \\ 2b^{2} + a & if x ≥ 2b. \end{matrix}

The graph of f_{c} is shown in Figure 15. We use f_{c} to construct a unit-length vector field X_{c}, defined in the strip S_{c} = 0 ≤ y ≤ 2b^{2} + 2a in R^{2}. Ultimately, we will stack these strips vertically, varying the parameter c, to obtain a vector field on R^{2}. The slope of the vector field X_{c} is the vertical linear interpolation between 0 on the lines x = 0 and x = 2b^{2} + 2a, and the unit tangent vector field to the graph of f_{c}. In other words,

X_{c}(x, y) = \frac{1}{\sqrt{1 + s_{c}(x, y)^{2}}} · (\frac{∂}{∂x} + s_{c}(x, y) · \frac{∂}{∂y}),

where:

s_{c}(x, y) = \{ \begin{matrix}f′_{c}(x)y/f_{c}(x) & if y ≤ f_{c}(x) \\ (2b^{2} + 2a − yf′_{c}(x)/(2b^{2} + 2a − f_{c}(x))) & if f_{c}(x) < y ≤ 2b^{2} + 2a. \end{matrix}

Figure 16: The vector field X_{c}.

Figure 16 shows X_{c}. Note that X_{c} is C^{1} and hence uniquely integrable in S_{c}.

Lemma 5.1 Given c = (a, b) as above, the function s_{c}(x, y) is Hölder continuous in the strip S_{c}, with exponent \frac{1}{2} and constant M, where M is independant of c.

Proof of Lemma 5.1: The strip S_{c} is naturally divided into four regions I-IV, as shown in in Figure 17. We first show that s_{c} is Hölder in a given (closed) region. By symmetry, it suffices to prove the claim for regions I and II.

Suppose (x, y) and (x, y′) are contained in the same region, either I or II (that is, 0 ≤ y, y′ ≤ f_{c}(x)). Note that |f′_{c}(x)| ≤ 2|f_{c}(x)|^{\frac{1}{2}} . It follows that

|s_{c}(x, y) − s_{c}(x, y′)| = |(y − y′)\frac{f′_{c}(x)}{f_{c}(x)}|
≤ |y − y′| · 2(f_{c}(x))^{− \frac{1}{2}}
≤ 2|y − y′|^{\frac{1}{2}}

Now suppose that (x, y) and (x′, y) are both contained in region I of S_{c}. An application of the Mean Value Theorem shows that

|s_{c}(x, y) − s_{c}(x′, y)| ≤ 2|x − x′|

Similarly, for (x, y) and (x′, y) in region II,

|s_{c}(x, y) − s_{c}(x′, y)| ≤ 6|x − x′|.

Figure 17: Regions I-IV in the strip S_{c}.

Hence s_{c} is 1/2 - Hölder continuous with constant 2 along vertical lines in a given region, and Lipschitz with constant 6 along horizontals. For sufficiently small values of a, b , the 1/2 - Hölder constant of s_{c} along horizontals is also 6. Finally, suppose that (x, y) and (x, y′) are in different regions. Then

|s_{c}(x, y) − s_{c}(x, y′)| ≤ |s_{c}(x, y) − s_{c}(x, f_{c}(x))| + |s_{c}(x, f_{c}(x)) − s_{c}(x, y′)|
≤ 6(|y − f_{c}(x)|^{\frac{1}{2}} + |f_{c}(x) − y′|^{\frac{1}{2}} )
≤ 6\sqrt{2}|y − y′|^{\frac{1}{2}}

Similarly, if (x, y), (x′, y) lie in different regions of S_{c}, then

|s_{c}(x, y) − s_{c}(x′, y)| ≤ 6\sqrt{2}|x − x′|^{\frac{1}{2}} .

Finally, for arbitrary (x, y), (x′, y′) in S_{c}, we have

|s_{c}(x, y) − s_{c}(x′, y′)| ≤ |s_{c}(x, y) − s_{c}(x, y′)| + |s_{c}(x, y′) − s_{c}(x′, y′)|
≤ 6\sqrt{2}(|y − y′|^{\frac{1}{2}} + |x − x′|^{\frac{1}{2}} )
≤ 6 · (2)^{\frac{3}{4}} d((x, y), (x′, y′))^{\frac{1}{2}}

Set M = 6 · (2)^{\frac{3}{4}} and the proof is complete. □

Hence the vector field X_{c} is Hölder-continuous, with Hölder constant M. Note that by modifying our construction of X_{c}, we could have replaced the Hölder exponent \frac{1}{2} by any fixed real number between 0 and 1.

We now proceed to the construction of the example.

Proposition 5.2 There is a vector field X on R^{2} that is 1/2-Hölder continuous, uniquely integrable, whose integral foliation fails to be α-Hölder continuous, for any α > 0.

Proof of Proposition 5.2:

For n ≥ 1, let c(n) = (2^{−n}, 2^{− \frac{\sqrt{n}}{2}} ). Let

h_{n} = ∑_{k=n+1}^{∞} 2(2^{−k} + 2^{−\sqrt{k}}),

h = ∑_{k=1}^{∞} 2(2^{−k} + 2^{−\sqrt{k}}),

To simplify notation, let X_{n}(x, y) = X_{c(n)}(x, y), s_{n} = s_{c(n)}, f_{n} = f_{c(n)}, and let S_{n} = S_{c(n)}. Define X as follows:

X(x, y) = \{ \begin{matrix}X_{n}(x, y − h_{n}) & if h_{n} ≤ y < h_{n+1}, for some n ≥ 1 \\ \frac{∂}{∂x} & otherwise. \end{matrix}

Note that the vertical width of the strip S_{n} is 2(2^{−n} + 2^{−\sqrt{n}}) so this defines a continuous vector field on R^{2}. Clearly X is uniquely integrable outside the closed region 0 ≤ y ≤ h. It is uniquely integrable in the region 0 < y ≤ h because it is uniquely integrable in every strip S_{n} + (0, h_{n}). Since the lines y = h_{n} are unique integral curves of X, the line y = 0 is also a unique integral curve. So X is uniquely integrable. Let F be the foliation of R^{2} tangent to X.

By Lemma 5.1, X is 1/2-Hölder-continuous in every strip S_{n}, with Hölder constant M (which is independent of n). To check that X is Hölder-continuous on all of R^{2}, it remains to estimate, for (x, y) ∈ S_{n} and (x′, y′) ∈ S_{n′} with n′ > n, the quantity:

|s_{n}(x, y) − s_{n′}(x′, y′)|.

Assume without loss of generality that s_{n}(x, y) ≥ s_{n′}(x′, y′). Then, since s_{n} ≥ 0, it follows that:

|s_{n}(x, y) − s_{n′}(x′, y′)| ≤ |s_{n}(x, y)|
= |s_{n}(x, y) − s_{n}(x′, h_{n})|
≤ M · d((x, y), (x′, h_{n}))^{\frac{1}{2}}
≤ M · d((x, y), (x′, y′))^{\frac{1}{2}} .

(A similar argument works when s_{n}(x, y) < s_{n′}(x′, y′)).

To see that F fails to be Hölder-continuous at the point (0, 0), first observe that the curves {(x, h_{n} + f_{n}(x)) | x ∈ R} are integral curves of X, as are the lines y = h_{n}. For δ > 0, there exists an N such that for n ≥ N,

sup_{x∈[−δ,δ]} |(h_{n} + f_{n}(x)) − h_{n}| = sup_{x∈R} f_{n}(x)
= 2 · 2^{−\sqrt{n}} + 2^{−n},

which is O(2^{−\sqrt{n}}). On the other hand, for all δ > 0 and n > 0,

inf_{x∈[−δ,δ]} |(h_{n} + f_{n}(x)) − h_{n}| = inf_{x∈R} f_{n}(x)
= 2^{−n},

which is O(2^{−n}). This implies that for every neighborhood U of the origin (0, 0) there exists N > 0 such that for every n ≥ N, there exist vertical transversals τ_{1} and τ_{2} and points z, z′, w, w′ ∈ U such that z, z′ ∈ τ_{1}, w = τ_{2} ∩ F(z), w′ = τ_{2} ∩ F(z′), and

d(w, w′) ≥ d(z, z′)^{\frac{1}{\sqrt{n}}} .

Clearly no F-holonomy map originating at the origin is α-Hölder-continuous, for any α > 0. □

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