On the ergodicity of partially hyperbolic systems

Keith Burns and Amie Wilkinson

June 11, 2008

Abstract

Pugh and Shub [PS3] have conjectured that essential accessibility implies ergodicity, for a C^{2}, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for C^{1+δ} partially hyperbolic systems.

Introduction

In [Ho] Eberhard Hopf introduced a simple argument that proved the ergodicity (with respect to Liouville measure) of the geodesic flow of a compact, negatively curved surface. The argument has since been applied to increasingly general classes of dynamical systems. The key feature that these systems possess is hyperbolicity. The strongest form of hyperbolicity is uniform hyperbolicity. A diffeomorphism f : M → M of a compact manifold M is uniformly hyperbolic or Anosov if there exists a splitting of the tangent bundle into T f-invariant subbundles:

T M = E^{s} ⊕ E^{u},

and a continuous Riemannian metric, such that for every unit vector v ∈ T M:

∥T fv∥ < 1  if v ∈ E^{s},  (1)
∥T fv∥ > 1 if v ∈ E^{u}.  (2)

Anosov flows are defined similarly, with E^{s}⊕E^{u} complementary to the bundle E^{0} that is tangent to the flow direction. The bundles E^{s} and E^{u} of an Anosov system are tangent to the stable and unstable foliations W^{s} and W^{u}, respectively. The properties of these foliations play a crucial role in the Hopf argument.

Hopf’s original argument established ergodicity for volume-preserving uniformly hyperbolic systems under the assumption that the foliations W^{s} and W^{u} are C^{1}. While the leaves of these foliations are always as smooth as the diffeomorphism, the foliations are usually only only Hölder continuous in the direction transverse to the leaves. In particular, for geodesic flows on arbitrary compact manifolds of negative sectional curvature, these foliations are not always C^{1}.

Anosov and Sinai [AS, A] observed that the C^{1} condition on the stable and unstable foliations in the Hopf argument could be replaced by the weaker condition of absolute continuity, which we discuss in Section 2.2. They showed that W^{s} and W^{u} are absolutely continuous if the system is C^{2}, thereby establishing ergodicity of all C^{2} volume-preserving uniformly hyperbolic systems, including geodesic flows for compact manifolds of negative sectional curvature.

At this point, it became clear that the Hopf argument should extend to even more general settings. Two natural generalizations of uniform hyperbolicity are:

• nonuniform hyperbolicity, which requires hyperbolicity along almost every orbit, but allows the expansion of E^{u} and the contraction of E^{s} to weaken near the exceptional set where there is no hyperbolicity; and

• partial hyperbolicity, which requires uniform expansion of E^{u} and uniform contraction of E^{s}, but allows central directions at each point, in which the expansion and contraction is dominated by the behavior in the hyperbolic directions.

The first direction is Pesin theory; the second is the subject of this paper.

Brin and Pesin [BP] and independently Pugh and Shub [PS1] first examined the ergodic properties of partially hyperbolic systems soon after the work of Anosov and Sinai. The current definition of partial hyperbolicity is more general than theirs, but has the same basic features.^{1} We say that a diffeomorphism f : M → M of a compact manifold M is partially hyperbolic if the following conditions hold. There is a nontrivial splitting of the tangent bundle, T M = E^{s} ⊕ E^{c} ⊕ E^{u}, that is invariant under the derivative map T f. Further, there is a Riemannian metric for which we can choose continuous positive functions ν, ν̂, γ and γ̂ with

1The difference is that in [BP] the functions ν, ν̂, γ and γ̂ in the definition of partial hyperbolicity are assumed to be constant.

ν, ν̂ < 1 and ν < γ < γ̂^{−1} < ν̂^{−1}  (3)

such that, for any unit vector v ∈ T_{p}M,

∥T fv∥ < ν(p),  if v ∈ E^{s}(p),  (4)
γ(p) < ∥T fv∥ < γ̂(p)^{−1},  if v ∈ E^{c}(p),  (5)
ν̂(p)^{−1} < ∥T fv∥,  if v ∈ E^{u}(p).  (6)

Partial hyperbolicity is a C^{1}-open condition: any diffeomorphism sufficiently C^{1}-close to a partially hyperbolic diffeomorphism is itself partially hyperbolic. Partially hyperbolic flows are defined similarly. For an extensive discussion of examples of partially hyperbolic dynamical systems, see the survey article [BPSW] and the book [P]. Among these examples are: the time-1 map of an Anosov flow, the frame flow for a compact manifold of negative sectional curvature, and many affine transformations of compact homogeneous spaces.

All of these examples preserve the volume induced by a Riemannian metric on M. The methods based on Hopf’s argument actually apply to a slightly larger class of invariant measures, namely those that lie in the measure class of a volume, meaning that they have the same sets of measure 0 as a volume (note that the volumes of any two Riemannian metrics lie in the same measure class). By a slight abuse of notation, we say that f is volume-preserving if it preserves a probability measure in the measure class of a volume. In this paper, m will denote a Riemannian volume on M (not necessarily invariant), and μ will denote an invariant probability measure that lies in the measure class of m. Measurability is with respect to the Borel σ-algebra on M.

As in the Anosov case, the stable and unstable bundles E^{s} and E^{u} of a partially hyperbolic diffeomorphism are tangent to foliations, which we again denote by W^{s} and W^{u} respectively [BP]. Brin-Pesin and Pugh-Shub proved that these foliations are absolutely continuous.

By its very nature, the Hopf argument shows that for almost every p ∈ M, almost every point of W^{s}(p) and almost every point of W^{u}(p) lies in the ergodic component of p. Thus we can only hope to prove ergodicity using a Hopf argument if something close to the following condition holds.

Definition: A partially hyperbolic diffeomorphism f : M → M is accessible if any point in M can be reached from any other along an su-path, which is a concatenation of finitely many subpaths, each of which lies entirely in a single leaf of W^{s} or a single leaf of W^{u}.

The accessibility class of p ∈ M is the set of all q ∈ M that can be reached from p along an su-path. Accessibility means that there is one accessibility class, which contains all points. The following notion is a natural weakening of accessibility.

Definition: A partially hyperbolic diffeomorphism f : M → M is essentially accessible if every measurable set that is a union of entire accessibility classes has either full or zero volume.

Pugh and Shub have conjectured that essential accessibility implies ergodicity, for a C^{2}, partially hyperbolic, volume-preserving diffeomorphism [PS2]. We prove this conjecture under one, rather mild additional assumption.

Definition: A partially hyperbolic diffeomorphism is center bunched if the functions ν, ν̂, γ, and γ̂ can be chosen so that:

ν < γγ̂ and ν̂ < γγ̂.  (7)

Our main result is:

Theorem 0.1 Let f be C^{2}, volume-preserving, partially hyperbolic and center bunched. If f is essentially accessible, then f is ergodic, and in fact has the Kolmogorov property.

This result extends earlier results about ergodicity of partially hyperbolic systems. Brin and Pesin [BP] proved in the early 1970’s that a C^{2} volumepreserving partially hyperbolic diffeomorphism that is essentially accessible is ergodic if it satisfies the following additional conditions:

• Center bunching: Inequalities (7) hold.

• Dynamical coherence: There are foliations W^{c}, W^{cs} and W^{cu} tangent to E^{c}, E^{c} ⊕ E^{s} and E^{c} ⊕ E^{u} respectively.

• Lipschitzness of W^{c}: There are Lipschitz foliation charts for W^{c}.

While the Brin-Pesin argument applies to many examples of partially hyperbolic diffeomorphisms, the third condition is in some ways very restrictive: Lipschitzness of W^{c} can be destroyed by arbitrarily small perturbations [SW2]. Brin and Pesin’s theorem applies in particular to the time-1 map φ_{1} of the geodesic flow for a compact surface of constant negative curvature. If we make a C^{1} small perturbation to φ_{1}, all of Brin and Pesin’s hypotheses continue to hold, except Lipschitzness of W^{c} (this fact follows from combining results in [D] with an argument of Mañé — see [BPSW], p. 352).

It was not until the 1990’s that Grayson, Pugh and Shub [GPS] were able to show that any small perturbation of φ_{1} is ergodic; in other words, φ_{1} is stably ergodic: any C^{2} volume-preserving diffeomorphism sufficiently C^{1}-close to φ_{1} is ergodic. The ideas in this groundbreaking paper have been generalized in several stages [W, PS2, PS3], culminating in [PS3]. The main result of [PS3] assumes dynamical coherence and uses a significantly stronger version of center bunching than inequalities (7). The center bunching hypothesis in [PS3] requires that the action of T f on E^{c} be close to isometric — that is, both γ and γ̂ (and not just their product) must be close to 1.

By contrast, our center bunching hypothesis requires only that the action of T f on E^{c} be close enough to conformal that the hyperbolicity of f dominates the nonconformality of T f on E^{c}. Center bunching always holds when T f|_{E^{c}} is conformal. For then we have ∥T_{p}fv∥ = ∥T_{p}f|_{E^{c}(p)}∥ for any unit vector v ∈ E^{c}(p), and hence we can choose γ(p) slightly smaller and γ̂(p)^{−1} slightly bigger than

∥T_{p}f|_{E^{c}(p)}∥.

By doing this we may make the ratio γ(p)/γ̂(p)^{−1} = γ(p)γ̂(p) arbitrarily close to 1, and hence larger than both ν(p) and ν̂(p).

In particular, center bunching holds whenever E^{c} is one-dimensional. As a corollary, we obtain:

Corollary 0.2 Let f be C^{2}, volume-preserving and partially hyperbolic with dim(E^{c}) = 1. If f is essentially accessible, then f is ergodic, and in fact has the Kolmogorov property.

This establishes the Pugh-Shub Conjecture mentioned above in the case where E^{c} is 1-dimensional.

Corollary 0.2 has also been recently proved by F. Rodríguez Hertz, J. Rodríguez Hertz, and R. Ures [HHU]. Their argument is mainly based on techniques in an earlier version^{2} of the present paper [BW1]. They prove in addition that stable accessibility is C^{r} dense among the C^{r} partially hyperbolic diffeomorphisms with 1-dimensional center, which implies, for r ≥ 2, that stable ergodicity is C^{r} dense among the volume-preserving C^{r} partially hyperbolic diffeomorphisms with one-dimensional center. Their work establishes the main stable ergodicity conjectures of Pugh and Shub ([PS3], Conjectures 1-3) in the case where E^{c} is one-dimensional.

There is only one place in the proof of Theorem 0.1 where we need the diffeomorphism to be C^{2} as opposed to C^{1+δ}, for some δ > 0. This is when we use the fact that center bunching implies that the stable and unstable holonomies between center leaves are Lipschitz. This fact is proved using a graph transform argument in [BP] and also (in a slightly more general setting) in [PSW, PSWc]. In [BW2], we show that the same result about holonomies holds when C^{2} is replaced by C^{1+δ}, at the expense of a more stringent bunching hypothesis. Plugging this result into the proof of Theorem 0.1, we obtain:

Theorem 0.3 Let f be C^{1+δ}, volume-preserving, and partially hyperbolic. Let μ, μ̂ be continuous functions satisfying:

μ(p) < ∥T fv∥,  if v ∈ E^{s}(p),  (8)
∥T fv∥ < μ̂(p)^{−1},  if v ∈ E^{u}(p).  (9)

Suppose that f satisfies the strong center bunching condition:

ν^{θ} < γγ̂  and  ν̂^{θ} < γγ̂,  (10)

where θ ∈ (0, δ) satisfies the inequalities:

νγ^{−1} < μ^{θ},  ν̂γ̂^{−1} < μ̂^{θ}.  (11)

If f is essentially accessible, then f is ergodic, and in fact has the Kolmogorov property.

2This earlier version proved the same result as the present paper but under the additional hypothesis of dynamical coherence, i.e. the existence of foliations tangent to the bundles E^{c} ⊕ E^{u} and E^{c} ⊕ E^{s}.

We remark that any θ satisfying the conditions in (11) is a Hölder exponent for the central distribution E^{c} (see, e.g., Theorem A in [PSW]). It would be interesting to know whether (10) could be replaced by (7) in Theorem 0.3. The strong center bunching condition in (10) is automatically satisfied when E^{c} is one-dimensional, since, as above, we may then choose γ and γ̂^{−1} so that γγ̂ is arbitrarily close to 1. As a corollary of Theorem 0.3, we obtain:

Corollary 0.4 Let f be C^{1+δ}, volume-preserving and partially hyperbolic with dim(E^{c}) = 1. If f is essentially accessible, then f is ergodic, and in fact has the Kolmogorov property.

Our arguments rely upon the same basic strategy as those of Pugh and Shub in [PS3]. We use a similar notion of center-unstable juliennes, and show that they are quasi-preserved by stable holonomies. But our analysis of the relationship between juliennes and density points is fundamentally different from theirs. One essential novelty is the use of a version of Cavalieri’s principle, introduced in Subsection 2.3. Another is the use of the fake foliations constructed in in Section 3. The use of Cavalieri’s principle greatly simplifies the analysis of the relationship between juliennes and Lebesgue density points of invariant sets, which is the heart of the proof. It allows us to weaken the center bunching hypothesis to the minimum condition needed to obtain smoothness of holonomy along stable leaves inside a leaf of a center stable foliation. The fake foliations and the simplifications using Cavalieri’s principle make it possible to eliminate the hypothesis of dynamical coherence. It is interesting to note that both advances, fake foliations and Cavalieri’s principle, are used in an essential way to remove the dynamical coherence assumption; the technique of fake foliations cannot be used directly to modify or improve the Pugh Shub approach to remove dynamical coherence.

Our ergodicity result, Theorem 0.1, is proved in Section 5 as a consequence of Theorem 5.1, which is really the central result of the paper. Theorem 5.1 is proved in Sections 6, 7 and 8.

We thank Marcelo Viana for useful comments and for telling us about the proof of absolute continuity of stable foliations in the pointwise partially hyperbolic setting in [AV]. We thank Charles Pugh and Mike Shub for very useful comments. Keith Burns was supported by NSF grants DMS-0100416 and DMS-0408704, and Amie Wilkinson by NSF grants DMS-0100314 and DMS-0401326. A proof of Theorem 0.1 under the additional hypothesis of dynamical coherence appeared earlier in the unpublished preprint [BW1].

1 Preliminaries

1.1 Notational conventions

We use the convention that if q is a point in M and j is an integer, then q_{j} denotes the point f^{j}(q), with q_{0} = q. If α : M → R is a positive function, and j ≥ 1 is an integer, let

α_{j}(p) = α(p)α(p_{1}) ⋯ α(p_{j−1}),

and

α_{−j}(p) = α(p_{−j})^{−1}α(p_{−j+1})^{−1} ⋯ α(p_{−1})^{−1}.

We set α_{0}(p) = 1. Observe that α_{j} is a multiplicative cocycle; in particular, we have α_{−j}(p)^{−1} = α_{j}(p_{−j}). Note also that (αβ)_{j} = α_{j}β_{j}, and if α is a constant function, then α_{n} = α^{n}.

The notation α < β, where α and β are continuous functions, means that the inequality holds pointwise, and the function min{α, β} takes the value min{α(p), β(p)} at the point p.

As usual P = O(Q) means that there is a constant C > 0 such that |P | ≤ CQ. Usually P and Q will depend on an integer n and one or more points in M. The constant C must be independent of n and the choice of the points.

1.2 Foliation boxes and local leaves

Let F be a foliation of an n-manifold M with d-dimensional smooth leaves. For r > 0, we denote by F(x, r) the connected component of x in the intersection of F(x) with the ball B(x, r).

A foliation box for F is the image U of R^{n−d}×R^{d} under a homeomorphism that sends each vertical R^{d}-slice into a leaf of F. The images of the vertical R^{d}-slices will be called local leaves of F in U.

A smooth transversal to F in U is a smooth codimension-d disk in U that intersects each local leaf in U exactly once and whose tangent bundle is uniformly transverse to T F. If τ_{1} and τ_{2} are two smooth transversals to F in U, we have the holonomy map h_{F} : τ_{1} → τ_{2}, which takes a point in τ_{1} to the intersection of its local leaf in U with τ_{2}.

1.3 Adapted metrics

We assume that the Riemannian metric on M is chosen so that the inequalities (3)–(6) involving ν, γ, ν̂, γ̂ in the Introduction hold. Such a metric will be called adapted. Note that a rescaling of an adapted metric is still adapted. It will be convenient to assume that the metric is scaled so that the geodesic balls of radius 1 are very small neighborhoods of their centers. Distance with respect to the metric will be denoted by d.

There is no harm in increasing ν and ν̂ and decreasing γ, γ̂ slightly, provided that the inequalities (3)–(6) still hold. If f is center bunched, the change must also be small enough so that inequalities (7) still hold. Similarly, if f is strongly center bunched, the change must also be small enough so that inequalities (10) still hold.

By rescaling the metric on M, we may assume that for some R > 1, and any x ∈ M, the Riemannian ball B(x, R) is contained in foliation boxes for both W^{s} and W^{u}. We assume that R is large enough so that all the objects considered in the sequel are small compared with R. Having fixed such an R, we define, for a = s or u, the local leaf of W^{a} through x by:

W_{loc}^{a}(x) = W^{a}(x, R).

Any foliation box U for either W^{s} or W^{u} that we consider in the rest of the paper will be small enough so that W_{loc}^{a}(x) ∩ U is a local leaf of W^{a} in U for each x ∈ U. By (if necessary) further rescaling the metric to make the local leaves smaller, we may assume that our metric is still adapted, and that for all p ∈ M, and q, q′ ∈ B(p, R), we have the following:

q ∈ W_{loc}^{s}(q′) ⟹ d(f(q), f(q′)) ≤ ν(p)d(q, q′),  (12)

and similarly,

q ∈ W_{loc}^{u}(q′) ⟹ d(f^{−1}(q), f^{−1}(q′)) ≤ ν̂(f^{−1}(p))d(q, q′).

In particular, f(W_{loc}^{s}(p)) ⊂ W_{loc}^{s}(f(p)) and f^{−1}(W_{loc}^{u}(p)) ⊂ W_{loc}^{u}(f^{−1}(p)), for all p ∈ M. This is possible because ν and ν̂ are continuous, and the inequalities (4) and (6) that they satisfy are strict.

An inductive argument then gives:

Lemma 1.1 If q_{j}, q′_{j} ∈ B(p_{j}, R) for j = 0, …, n − 1, and q ∈ W_{loc}^{s}(q′), then

d(q_{n}, q′_{n}) ≤ ν_{n}(p)d(q, q′).

If q_{−j}, q′_{−j} ∈ B(p_{−j}, R) for j = 0, …, n − 1, and q ∈ W_{loc}^{u}(q′), then

d(q_{−n}, q′_{−n}) ≤ ν̂_{−n}(p)^{−1}d(q, q′).

Proof. We prove the first claim; the second follows from the first, with f replaced by f^{−1}. The proof is by induction on n. The claim is vacuously true for n = 0. Suppose the claim holds for n = k. The inductive assumption gives that

d(q_{k}, q′_{k}) ≤ ν_{k}(p)d(q, q′)

and q′_{k} ∈ W_{loc}^{s}(q_{k}). Then (12), applied at p_{k}, implies that q′_{k+1} ∈ W_{loc}^{s}(q_{k+1}), and

d(q_{k+1}, q′_{k+1}) ≤ ν(p_{k})d(q_{k}, q′_{k})
≤ ν(p_{k})ν_{k}(p)d(q, q′)
= ν(p_{k+1})d(q, q′).

⋄

2 Density points and absolute continuity

2.1 Volume and density

Recall that when we say that the diffeomorphism f is volume-preserving, we mean that f preserves a probability measure μ that lies in the measure class of a Riemannian volume m on M.

If S ⊆ M is a smooth submanifold, we denote by m_{S} the volume of the induced Riemannian metric on S. If F is a foliation with smooth leaves, and A is contained in a single leaf of F and is measurable in that leaf, then we denote by m_{F}(A) the induced Riemannian volume of A in that leaf. A set is said to be saturated by a foliation F or F-saturated if it is a union of entire leaves of F. A set A is essentially F-saturated if there exists a measurable F-saturated set A′, which we call an essential F-saturate of A, with m(A Δ A′) = 0.

If ν is a measure and A and B are ν-measurable sets with ν(B) > 0, we define the density of A in B by:

ν(A : B) = \frac{ν(A ∩ B)}{ν(B)}.

A point x ∈ M is a Lebesgue density point of a measurable set X ⊆ M if

lim_{r→0} m(X : B_{r}(x)) = 1.

Notice that the notion of Lebesgue density point depends only on the smooth structure of M, because any two Riemannian metrics have the same Lebesgue density points.

The Lebesgue Density Theorem implies that if A is a measurable set and Â is the set of Lebesgue density points of A, then m(A Δ Â) = 0.

Lebesgue density points can be characterized using nested sequences of measurable sets. We say that a sequence of measurable sets Y_{n} nests at a point x if Y_{0} ⊃ Y_{1} ⊃ Y_{2} ⊃ ⋯ ⊃ {x}, and

⋂_{n} Y_{n} = {x}.

A sequence Y_{n}, with m(Y_{n}) > 0, that nests at x is a Lebesgue density sequence at x if for every measurable set X, x is a Lebesgue density point of X if and only if:

lim_{n→∞} m(X : Y_{n}) = 1.

It is easily shown that a Lebesgue density sequence Y_{n} must be regular, a term we now define.

Definition: A nested sequence of measurable sets Y_{n} is regular if there exists δ > 0 such that, for all n ≥ 0, we have m(Y_{n}) > 0, and

m(Y_{n+1}) ≥ δm(Y_{n}).

The simplest example of a Lebesgue density sequence at x is the sequence of balls B(x, ρ^{n}), where ρ ∈ (0, 1).

In our proof of Theorem 0.1, we characterize the Lebesgue density points of a special class of measurable sets, those that are both essentially W^{s}- saturated and essentially W^{u}-saturated. Such sets will be called bi essentially saturated. These sets arise when we consider Birkhoff averages of continuous test functions; the sublevel sets of such averages are bi essentially saturated.

We say that Y_{n} is a Lebesgue density sequence at x for bi essentially saturated sets if Y_{n} nests at x, Y_{n} is regular, and, for every bi essentially saturated set X, x is a Lebesgue density point of X if and only if:

lim_{n→∞} m(X : Y_{n}) = 1.

Note that Lebesgue density sequence for bi essentially saturated sets is not necessarily a Lebesgue density sequence, because only certain measurable sets are considered. In fact, many of the Lebesgue density sequences for bi essentially saturated sets constructed in this paper are not Lebesgue density sequences.

In our proof we will frequently have to pass the property of being a Lebesgue density sequence for bi essentially saturated sets from one sequence Y_{n} that nests at x to another sequence Z_{n} that nests at x. In order to do so, we have to show that Z_{n} is also regular and that, for every measurable set X that is bi essentially saturated,

lim_{n→∞} m(X : Y_{n}) = 1 ⟺ lim_{n→∞} m(X : Z_{n}) = 1.  (13)

We have two techniques for doing this. Which technique we use depends on the construction of Y_{n} and Z_{n}.

The first technique is very simple. Two nested sequences of sets Y_{n} and Z_{n} are internested if there exists a k ≥ 1 such that, for all n ≥ 0, we have

Y_{n+k} ⊆ Z_{n},  and  Z_{n+k} ⊆ Y_{n}.

Internestedness is an equivalence relation. The following lemma is a straightforward consequence of the definitions.

Lemma 2.1 Let Y_{n} and Z_{n} be internested sequences of measurable sets, with Y_{n} regular. Then Z_{n} is also regular. If the sets Y_{n} have positive measure, then so do the Z_{n}, and, for any measurable set X,

lim_{n→∞} m(X : Y_{n}) = 1 ⟺ lim_{n→∞} m(X : Z_{n}) = 1.

Corollary 2.2 Suppose that Y_{n} and Z_{n} both nest at x and are internested. Then Y_{n} is a Lebesgue density sequence for bi essentially saturated sets if and only if Z_{n} is a Lebesgue density sequence for bi essentially saturated sets.

The second technique uses absolute continuity of the foliations W^{s} and W^{u}, plus the saturation properties of X, and is developed in the next two subsections.

2.2 Absolute continuity

Our arguments in this paper use two versions of the property of absolute continuity of a foliation.

The first version of absolute continuity involves holonomy maps between transversals. A foliation F with smooth leaves is transversely absolutely continuous with bounded Jacobians if for every angle α ∈ (0, π/2], there exists C ≥ 1 and R_{0} > 0 such that, for every foliation box U of diameter less than R_{0}, any two smooth transversals τ_{1}, τ_{2} to F in U of angle at least α with F, and any m_{τ_{1}}–measurable set A contained in τ_{1}:

C^{−1}m_{τ_{1}}(A) ≤ m_{τ_{2}}(h_{F}(A)) ≤ Cm_{τ_{1}}(A).  (14)

The second version involves a Fubini-like property. A foliation F with smooth leaves is absolutely continuous with bounded Jacobians if, for every α ∈ (0, π/2], there exist C ≥ 1 and R_{0} > 0 such that, for every foliation box U of diameter less than R_{0}, any smooth transversal τ to F in U of angle at least α with F, and any measurable set A contained in U, we have the inequality:

C^{−1}m(A) ≤ ∫_{τ} m_{F}(A ∩ F_{loc}(x)) dm_{τ}(x) ≤ Cm(A).  (15)

It is not difficult to verify that these two properties are equivalent. Note that the minimal C for which (14) holds is not necessarily the same as the minimal C for which (15) holds.

The foliations W^{s} and W^{u} for a partially hyperbolic diffeomorphism are transversely absolutely continuous with bounded Jacobians. This was shown in the Anosov case by Anosov [A], and in the case of partial hyperbolicity by Brin-Pesin and Pugh-Shub [BP, PS1]. Their proofs were written under the assumption that the function ν, ν̂, γ and γ̂ are constant. In the general case of partial hyperbolicity, where these functions are not constant, absolute continuity of W^{s} and W^{u} follows from Pesin theory. A direct proof in this context has been given by Abdenur and Viana [AV]. All of these results show that the Jacobians are continuous functions, and so are bounded, since M is compact. In general, W^{c} does not have either absolute continuity property, even when f is dynamically coherent (examples were first constructed by Katok [Mi]; open sets of examples by Shub-Wilkinson [SW2]).

2.3 Saturated sets and Cavalieri’s principle

The second technique mentioned in subsection 2.1 involves decomposing the sets in a sequence nesting at x along the leaves of an absolutely continuous foliation. The method is reminiscent of Cavalieri’s Principle, which states that the volume of a 3-dimensional solid may computed by taking 2-dimensional slices of the solid. If the areas of these slices are approximately equal, then the volume of the solid is approximately equal to the product of the slice area and the total height of the the slices.

In our setting, we use the leaves of an absolutely continuous foliation to decompose a subset of a foliation box into slices, which we call fibers. The idea of approximately equal area of the slices translates into c-uniformity of the fibers, which we define below. When the foliation is absolutely continuous, the volume of a subset of the foliation box with c-uniform fibers will be approximately the volume of one of the fibers times the volume of the projection of the set onto a transversal.

Let F be an absolutely continuous foliation and let U be a foliation box for F. Let τ be a smooth transversal to F in U. Let Y ⊆ U be a measurable set. For a point q ∈ τ, we define the fiber Y (q) of Y over q to be the intersection of Y with the local leaf of F in U containing q. The base τ_{Y} of Y is the set of all q ∈ τ such that the fiber Y (q) is m_{F}-measurable and m_{F}(Y (q)) > 0. The absolute continuity of F implies that τ_{Y} is m_{τ}-measurable. We say “Y fibers over Z” to indicate that Z = τ_{Y}.

If, for some c ≥ 1, the inequalities

c^{−1} ≤ \frac{m_{F}(Y (q))}{m_{F}(Y (q′))} ≤ c

hold for all q, q′ ∈ τ_{Y}, then we say that Y has c-uniform fibers. A sequence of measurable sets Y_{n} contained in U has c-uniform fibers if each set in the sequence has c-uniform fibers, with c independent of n.

Proposition 2.3 Suppose that the foliation F is absolutely continuous with bounded Jacobians. Let U be a foliation box for F, and let τ be a smooth transversal to F in U. Then there is a constant C ≥ 1 such that for any c ≥ 1, any measurable set Y ⊂ U with c-uniform fibers, and any point q_{0} ∈ τ_{Y}, we have

(Cc)^{−1}m(Y) ≤ m_{F}(Y (q_{0}))m_{τ}(τ_{Y}) ≤ Ccm(Y).

Proof. Absolute continuity of F with bounded Jacobians implies that there exists a C ≥ 1, that depends only on F, U, and τ, and does not depend on Y, such that

C^{−1}m(Y) ≤ ∫_{τ} m_{F}(Y (q)) dm_{τ}(q) ≤ Cm(Y).

Since the fibers of Y are c-uniform, we have

c^{−1}m_{F}(Y (q_{0})) ≤ m_{F}(Y (q)) ≤ cm_{F}(Y (q_{0})),

for any q ∈ τ_{Y}. Combing these two sets of inequalities gives the desired result. ⋄

Regularity of a sequence with c-uniform fibers can be obtained from regularity of its fibers and bases.

Proposition 2.4 Suppose that the foliation F is absolutely continuous with bounded Jacobians. Let U be a foliation box for F, and let τ be a smooth transversal to F in U. Let Y_{n} be a sequence of subsets of U with c-uniform fibers. Suppose that there exists δ > 0 such that:

1. for all n ≥ 0,

m_{τ}(τ_{Y_{n+1}}) ≥ δm_{τ}(τ_{Y_{n}});

2. for all n ≥ 0, there are points z ∈ τ_{Y_{n+1}}, z′ ∈ τ_{Y_{n}} with

m_{F}(Y_{n+1}(z)) ≥ δm_{F}(Y_{n}(z′)).

Then Y_{n} is regular.

Proof. It follows from Proposition 2.3 that

m(Y_{n+1}) ≥ (cC)^{−1}m_{F}(Y_{n+1}(z))m_{τ}(τ_{Y_{n+1}}),

and

m(Y_{n}) ≤ cCm_{F}(Y_{n}(z′))m_{τ}(τ_{Y_{n}}).

Using the two properties of δ, we then obtain

m(Y_{n+1}) ≥ (cC)^{−2}δ^{2}m(Y_{n}),

which says that Y_{n} is regular. ⋄

We now turn to the second technique mentioned above for proving an equivalence of the form (13). The main result we prove in this section is:

Proposition 2.5 Let F be absolutely continuous with bounded Jacobians, and let U be a foliation box for F with smooth transversal τ. Let Y_{n} and Z_{n} be sequences of measurable subsets of U with c-uniform fibers, for some c ≥ 1. Suppose that τ_{Y_{n}} = τ_{Z_{n}}, for all n. Then, for any essentially F-saturated set X ⊆ U, we have the equivalence:

lim_{n→∞} m(X : Y_{n}) = 1 ⟺ lim_{n→∞} m(X : Z_{n}) = 1.

Corollary 2.6 Fix F = W^{s} or W^{u}. Suppose that Y_{n} and Z_{n} satisfy the hypotheses of Proposition 2.5, and that Y_{n} and Z_{n} both nest at x. If Y_{n} is a Lebesgue density sequence at x for bi essentially saturated sets, and Z_{n} is regular, then Z_{n} is a Lebesgue density sequence at x for bi essentially saturated sets.

Before proving Proposition 2.5, we establish a related result, which will also be used in the proof of Theorem 0.1.

Proposition 2.7 Let F be absolutely continuous with bounded Jacobians, and let U be a foliation box for F with smooth transversal τ. Suppose that {Y_{n}}_{n≥0} is a sequence of measurable sets in U with c-uniform fibers, for some c ≥ 1. Then, for every F-saturated measurable set X, we have the equivalence:

lim_{n→∞} m(X : Y_{n}) = 1 ⟺ lim_{n→∞} m_{τ}(τ_{X} : τ_{Y_{n}}) = 1.

Remark: The hypothesis that X is F-saturated can be weakened: it suffices for X ∩ U to be a union of local leaves of F in U.

Proof of Proposition 2.7. Let X^{∗} be the complement of X in M. Then X^{∗} is also F-saturated. The proposition can be reformulated in terms of X^{∗}. We have to prove the equivalence:

lim_{n→∞} m(X^{∗} : Y_{n}) = 0 ⟺ lim_{n→∞} m_{τ}(τ_{X^{∗}} : τ_{Y_{n}}) = 0.

For each n, let

m_{n} = inf_{q∈τ_{Yn}} m_{F}(Y_{n}(q)).

Since the fibers of Y_{n} are c-uniform, it follows that m_{n} > 0 for all n, and:

m_{n} ≤ m_{F}(Y_{n}(q)) ≤ cm_{n},

for all q ∈ τ_{Y_{n}}. Absolute continuity implies that there exists a C ≥ 1 such that

C^{−1}m(Y_{n}) ≤ ∫_{τ} m_{F}(Y_{n}(q)) dm_{τ}(q) ≤ Cm(Y_{n}).

Together, these inequalities imply that

C^{−1}m_{n}m_{τ}(τ_{Y_{n}}) ≤ m(Y_{n}) ≤ Ccm_{n}m_{τ}(τ_{Y_{n}}).  (16)

Since X^{∗} is F-saturated, the F-fiber of X^{∗} ∩ Y_{n} over a point q ∈ τ_{Y_{n}} is either empty or equal to Y_{n}(q). Thus X^{∗} ∩ Y_{n} also has c-uniform fibers, and, as above, we obtain:

C^{−1}m_{n}m_{τ}(τ_{X^{∗}∩Y_{n}}) ≤ m(X^{∗} ∩ Y_{n}) ≤ Ccm_{n}m_{τ}(τ_{X^{∗}∩Y_{n}}).  (17)

Noting that τ_{X^{∗}∩Y_{n}} = τ_{X^{∗}} ∩ τ_{Y_{n}} and dividing the inequalities in (17) by those in (16), we obtain:

(C^{2}c)^{−1}m_{τ}(τ_{X^{∗}} : τ_{Y_{n}}) ≤ m(X^{∗} : Y_{n}) ≤ C^{2}cm_{τ}(τ_{X^{∗}} : τ_{Y_{n}}).

The result follows easily from this. ⋄

Proof of Proposition 2.5. Let X′ be an essential F-saturate of X. Using Proposition 2.7, we have the equivalences:

lim_{n→∞} m(X : Y_{n}) = 1 ⟺ lim_{n→∞} m(X′ : Y_{n}) = 1
⟺ lim_{n→∞} m_{τ}(τ_{X′} : τ_{Y_{n}}) = 1
⟺ lim_{n→∞} m_{τ}(τ_{X′} : τ_{Z_{n}}) = 1
⟺ lim_{n→∞} m(X′ : Z_{n}) = 1
⟺ lim_{n→∞} m(X : Z_{n}) = 1. ⋄

3 Fake invariant foliations

The Lebesgue density sequences for bi essentially saturated sets that we use in this proof will be constructed using dynamical foliations with uniform continuity properties. If f happens to be dynamically coherent, then we are free to use the foliations W^{s}, W^{u}, W^{c}, W^{cs}, and W^{cu} for these constructions.

Since we are not assuming dynamical coherence, we must find substitutes for W^{c}, W^{cs}, and W^{cu} to make our proof work in general. It turns out to be simplest to find substitutes for all invariant foliations W^{s}, W^{u}, W^{c}, W^{cs}, and W^{cu}. We call these substitutes “fake invariant foliations.” There are a few key places in the argument where we will have to use the real invariant foliations W^{u} and W^{s}, rather than their fake counterparts. We will indicate where this is the case. The reader should recall the choice of the constant R from Section 1.3.

Proposition 3.1 Let f : M → M be a C^{1} partially hyperbolic diffeomorphism. For any ε > 0, there exist constants r and r_{1} with R > r > r_{1} > 0 such that, for every p ∈ M, the neighborhood B(p, r) is foliated by foliations Ŵ_{p}^{u}, Ŵ_{p}^{s}, Ŵ_{p}^{c}, Ŵ_{p}^{cu} and Ŵ_{p}^{cs} with the following properties, for each β ∈ {u, s, c, cu, cs}:

1. Almost tangency to invariant distributions: For each q ∈ B(p, r), the leaf Ŵ_{p}^{β}(q) is C^{1} and the tangent space T_{q}Ŵ_{p}^{β}(q) lies in a cone of radius ε about E^{β}(q).

2. Local invariance: for each q ∈ B(p, r_{1}),

f(Ŵ_{p}^{β}(q, r_{1})) ⊂ Ŵ_{f(p)}^{β}(f(q)), and f^{−1}(Ŵ_{p}^{β}(q, r_{1})) ⊂ Ŵ_{f^{−1}(p)}^{β}(f^{−1}(q)).

3. Exponential growth bounds at local scales: The following hold for all n ≥ 0.

(a) Suppose that q_{j} ∈ B(p_{j}, r_{1}) for 0 ≤ j ≤ n − 1. If q′ ∈ Ŵ_{p}^{s}(q, r_{1}), then q′_{n} ∈ Ŵ_{p}^{s}(q_{n}, r_{1}), and

d(q_{n}, q′_{n}) ≤ ν_{n}(p)d(q, q′).

If q′_{j} ∈ Ŵ_{p}^{cs}(q_{j}, r_{1}) for 0 ≤ j ≤ n − 1, then q′_{n} ∈ Ŵ_{p}^{cs}(q_{n}), and

d(q_{n}, q′_{n}) ≤ γ̂_{n}(p)^{−1}d(q, q′).

(b) Suppose that q_{−j} ∈ B(p_{−j}, r_{1}) for 0 ≤ j ≤ n − 1. If q′ ∈ Ŵ_{p}^{u}(q, r_{1}), then q′_{−n} ∈ Ŵ_{p}^{u}(q_{−n}, r_{1}), and

d(q_{−n}, q′_{−n}) ≤ ν̂_{−n}(p)^{−1}d(q, q′).

If q′_{−j} ∈ Ŵ_{p}^{cu}(q_{−j}, r_{1}) for 0 ≤ j ≤ n − 1, then q′_{−n} ∈ Ŵ_{p}^{cu}(q_{−n}), and

d(q_{−n}, q′_{−n}) ≤ γ_{−n}(p)d(q, q′).

4. Coherence: Ŵ_{p}^{s} and Ŵ_{p}^{c} subfoliate Ŵ_{p}^{cs}; Ŵ_{p}^{u} and Ŵ_{p}^{c} subfoliate Ŵ_{p}^{cu}.

5. Uniqueness: Ŵ_{p}^{s}(p) = W^{s}(p, r), and Ŵ_{p}^{u}(p) = W^{u}(p, r).

6. Regularity: If f is C^{1+δ}, then the foliations Ŵ_{p}^{u}, Ŵ_{p}^{s}, Ŵ_{p}^{c}, Ŵ_{p}^{cu} and Ŵ_{p}^{cs} and their tangent distributions are uniformly Hölder continuous.

7. Regularity of the strong foliation inside weak leaves: If f is C^{2} and center bunched, then each leaf of Ŵ_{p}^{cs} is C^{1} foliated by leaves of the foliation Ŵ_{p}^{s}, and each leaf of Ŵ_{p}^{cu} is C^{1} foliated by leaves of the foliation Ŵ_{p}^{u}. If f is C^{1+δ} and strongly center bunched, then the same conclusion holds.

The regularity statements in 6. and 7. hold uniformly in p ∈ M.

Proof. Suppose that f is C^{k}, for some k ≥ 1. After possibly reducing ε, we can assume that inequalities (3)–(6) hold for unit vectors in the ε-cones around the spaces in the partially hyperbolic splitting.

The construction of the leaves of Ŵ_{p}^{cu} and Ŵ_{p}^{cs} through p is essentially the same as the proof of the existence of pseudo-hyperbolic plaque families in [HPS]. They are obtained as fixed points of graph transforms of a map that coincides with f in a neighborhood of the orbit of p. We take the argument one step further and consider all fixed points of these graph transforms in the entire neighborhood of p.

Our construction will be performed in two steps. In the first, we construct foliations of each tangent space T_{p}M. In the second step, we use the exponential map exp_{p} to project these foliations from a neighborhood of the origin in T_{p}M to a neighborhood of p.

Step 1. We choose an r_{0} > 0 such that exp_{p}^{−1} is defined on B(p, 2r_{0}). For r ∈ (0, r_{0}], we define, in the standard way, a map:

[figure: T M; T M; M; M; F_{r}; f]

which is uniformly C^{k} on fibers, satisfying:

1. F_{r}(p, v) = exp_{f(p)}^{−1} ∘f ∘ exp_{p}(v), for ∥v∥ ≤ r; 2. F_{r}(p, v) = T_{p}f(v), for ∥v∥ ≥ 2r;

3. ∥F_{r}(p, ·) − T_{p}f(·)∥_{C^{1}} → 0 as r → 0, uniformly in p.

Endowing M with the discrete topology, we regard T M as the disjoint union of its fibers. Property 3. implies that, if r is small enough, then F_{r} is partially hyperbolic, and each bundle in the partially hyperbolic splitting for F_{r} at v ∈ T_{p}M lies within the ε/2-cone about the corresponding subspace of T_{p}M in the partially hyperbolic splitting for f at p (we are making the usual identification of T_{v}T_{p}M with T_{p}M). If r is small enough, the equivalents of inequalities (3)–(6) will hold with T f replaced by T F_{r}.

If r is sufficiently small, standard graph transform arguments give stable, unstable, center-stable, and center-unstable foliations for F_{r} inside each T_{p}M. These foliations are uniquely determined by the extension F_{r} and the requirement that their leaves be graphs of bounded functions. We obtain a center foliation by intersecting the leaves of the center-unstable and centerstable foliations. While T M is not compact, all of the relevant estimates for F_{r} are uniform, and it is this, not compactness, that counts. The proof of the Hadamard-Perron Theorem in [KH] contains many of the details of this argument.

The uniqueness of the stable and unstable foliations implies, via a standard argument (see, e.g. [HPS], Theorem 6.1 (e)), that the stable foliation subfoliates the center-stable, and the unstable subfoliates the center-unstable.

We now discuss the regularity properties of these foliations of T M. Recall the standard method for determining the regularity of invariant bundles and foliations. Suppose that T X = E_{1} ⊕ E_{2} is a T g-invariant splitting of the tangent bundle for a C^{k} diffeomorphism g : X → X satisfying, for every p ∈ X, and every unit vector v ∈ T_{p}X:

α_{1}(p) < ∥T gv∥ < β_{1}(p), if v ∈ E_{1}(p),  (18)
α_{2}(p) < ∥T gv∥  if v ∈ E_{2}(p),  (19)

where 0 < α_{1}(p) < β_{1}(p) < α_{2}(p). Then the bundle E_{2} is C^{a}, for any a ≤ k − 1 that satisfies:

sup_{p∈M} \frac{β_{1}(p)}{α_{1}(p)^{a}α_{2}(p)} < 1.

When the functions α_{1}, α_{2}, β_{2} are constant, this fact is classical — see, e.g. the C^{r} Section Theorem in [HPS]. The general case of this result appears more recently in the literature [SS, Ha, W, PSW]. This result extends, at least in part, to give regularity of invariant foliations. In particular, when there is a foliation F_{2} tangent to E_{2} that arises at the unique fixed point of a nonlinear graph transform, then F_{2} is a C^{a} foliation [PSW, PSWc]. These results are proved in the compact case, but compactness is used only to obtain uniform estimates on the functions α_{1}, β_{1}, α_{2} and the derivative of g; the results carry over as long as such uniform estimates hold.

Our foliations of T M have been constructed as the unique fixed points of graph transform maps. We can apply the above results to the F_{r}-invariant splittings of T T M as the sum of the stable and center-unstable bundles for F_{r} and as the sum of the center-stable and unstable bundles for F_{r}. It follows immediately that both the center-unstable and unstable bundles and the corresponding foliations are Hölder continuous as long as F_{r} is C^{1+δ} for some δ > 0. We obtain the Hölder continuity of the center-stable and stable bundles for F_{r} and the corresponding foliations by thinking of the same splittings as F_{r}^{−1}-invariant. Hölder regularity of the center bundle and foliation is obtained by noticing the center is the intersection of the centerstable and center-unstable.

When k ≥ 2, a similar estimate gives the C^{1} regularity of the unstable bundle along the leaves of the center-unstable foliation. The manifold X is the disjoint union of the leaves of the center-unstable foliation for F_{r}, E_{2} is the unstable bundle, and E_{1} is the center bundle. We have:

α_{1} = γ, β_{1} = γ̂^{−1}, and α_{2} = ν̂^{−1}.

The center bunching hypothesis ν̂ < γγ̂ implies that

sup_{p∈M} \frac{ν̂(p)}{γ(p)^{a} γ̂(p)} < 1,

for some a > 1. It follows that E_{2} = E^{u} is a C^{1} bundle over X, the leaves of the center-unstable foliation. Similarly, this argument shows that the center bunching hypothesis ν < γγ̂ implies that E^{s} is a C^{1} bundle over the leaves of the center-stable foliation. There is an additional difficulty in the proof of this estimate, which is that X is not a C^{k} manifold in general, even when F_{r} is C^{k}. This difficulty is dealt with in [PSW, PSWc].

When 1 < k < 2, this type of estimate does not work at all. A different argument, working with holonomy maps instead of bundles, is presented in [BW2]. The main result there implies that under the strong center bunching hypothesis on f, which carries over to F_{r}, the unstable foliation C^{1}-subfoliates the center-unstable, and the stable foliation C^{1}-subfoliates the center-stable.

Step 2. We now have foliations of T_{p}M, for each p ∈ M. We obtain the foliations Ŵ_{p}^{u}, Ŵ_{p}^{c}, Ŵ_{p}^{s}, Ŵ_{p}^{cu}, and Ŵ_{p}^{cs} by applying the exponential map exp_{p} to the corresponding foliations of T_{p}M inside the ball around the origin of radius r.

If r is sufficiently small, then the distribution E^{β}(q) lies within the angular ε/2-cone about the parallel translate of E^{β}(p), for every β ∈ {u, s, c, cu, cs} and all p, q with d(p, q) ≤ r. Combining this fact with the preceding discussion, we obtain that property 1. holds if r is sufficiently small.

Property 2. — local invariance — follows from invariance under F_{r} of the foliations of T M and the fact that exp_{f(p)}(F_{r}(p, v)) = f(exp_{p}(p, v)) provided ∥v∥ ≤ r.

Having chosen r, we now choose r_{1} small enough so that f(B(p, 2r_{1})) ⊂ B(f(p), r) and f^{−1}(B(p, 2r_{1})) ⊂ B(f^{−1}(p), r), and so that, for all q ∈ B(p, r_{1}),

q′ ∈ Ŵ_{p}^{s}(q, r_{1}) ⟹ d(f(q), f(q′)) ≤ ν(p) d(q, q′),
q′ ∈ Ŵ_{p}^{u}(q, r_{1}) ⟹ d(f^{−1}(q), f^{−1}(q′)) ≤ ν̂(f^{−1}(p)) d(q, q′),
q′ ∈ Ŵ_{p}^{cs}(q, r_{1}) ⟹ d(f(q), f(q′)) ≤ γ̂(p)^{−1} d(q, q′),  and
q′ ∈ Ŵ_{p}^{cu}(q, r_{1}) ⟹ d(f^{−1}(q), f^{−1}(q′)) ≤ γ(f^{−1}(p))^{−1} d(q, q′).

Property 3. — exponential growth bounds at local scales — is now proved by an inductive argument similar to the proof of Lemma 1.1.

Properties 4.– 7. — coherence, uniqueness, regularity and regularity of the strong foliation inside weak leaves — follow immediately from the corresponding properties of the foliations of T M discussed above. ⋄

Remark: Note that the system of local foliations constructed in Proposition 3.1 is not unique; it depends on the extension of F_{r} outside of a neighborhood of the zero-section of T M. Also note that, even when f is dynamically coherent, in general there is no reason to expect the fake invariant foliations Ŵ_{p}^{cs}, Ŵ_{p}^{cu}, and Ŵ_{p}^{c} to coincide with the local leaves of the real invariant foliations W^{cs}, W_{p}^{cu}, and W_{p}^{c}, even at p. For the leaf Ŵ_{p}^{cs}(p) to coincide with the local leaf of W^{cs} through p, it is necessary that every iterate f^{−n}(W^{cs}(p_{n}, r) overflow the neighborhood B(p, r).

For the rest of the paper, Ŵ_{p}^{s}, Ŵ_{p}^{c}, Ŵ_{p}^{s}, Ŵ_{p}^{cs} and Ŵ_{p}^{cu} will denote fake invariant foliations given by Proposition 3.1, with ε > 0 much less than the angle between any two of the subspaces in the partially hyperbolic splitting. We may rescale the metric so that the radius r_{1} in conclusion 2. of Proposition 3.1 is much bigger than 1. This will ensure that all of the objects used in the rest of the paper are well-defined. We may assume that if max{d(x, p), d(y, p)} ≤ 3, then Ŵ_{p}^{cs}(x) ∩ Ŵ_{p}^{u}(y), Ŵ_{p}^{cs}(x) ∩ W_{loc}^{u}(y), Ŵ_{p}^{cu}(x) ∩ Ŵ_{p}^{s}(y) and Ŵ_{p}^{cu}(x) ∩ W_{loc}^{s}(y) are single points. We denote by m̂_{a} the measure m_{W^{a}}.

4 Distortion estimates inside thin neighborhoods

4.1 A simple distortion lemma

The next lemma will be used to compare values of Hölder cocycles at nearby points.

Lemma 4.1 Let α : M → R be a positive Hölder continuous function, with exponent θ > 0. Then there exists a constant H > 0 such that the following holds, for all p, q ∈ M, B > 0 and n ≥ 1:

∑_{i=0}^{n−1} d(p_{i}, q_{i})^{θ} ≤ B ⟹ e^{−HB} ≤ \frac{α_{n}(p)}{α_{n}(q)} ≤ e^{HB},

and

∑_{i=1}^{n} d(p_{−i}, q_{−i})^{θ} ≤ B ⟹ e^{−HB} ≤ \frac{α_{−n}(p)}{α_{−n}(q)} ≤ e^{HB}.

Proof. We prove the first part of the lemma. The second part is proved similarly. The function log α is also Hölder continuous with exponent θ. Let H > 0 be the Hölder constant of log α, so that for all x, y ∈ M:

|log α(x) − log α(y)| ≤ Hd(x, y)^{θ}.

The desired inequalities are equivalent to:

|log α_{n}(p) − log α_{n}(q)| ≤ HB.

Expanding log α_{n} as a series, we obtain:

|log α_{n}(p) − log α_{n}(q)| ≤ ∑_{i=0}^{n−1} |log α(p_{i}) − log α(q_{i})|
≤ H ∑_{i=0}^{n−1} d(p_{i}, q_{i})^{θ}.
≤ HB,

since

∑_{i=0}^{n−1} d(p_{i}, q_{i})^{θ} ≤ B,

by the hypothesis of the lemma. ⋄

4.2 Thin neighborhoods of W^{s}(p, 1)

We next identify, for each n ≥ 0 and p ∈ M, a neighborhood of p whose first n iterates remain in a uniform neighborhood of the corresponding iterates of p. We give an exponential estimate of the size of the first n iterates of the nth such neighborhood. In our proof of Theorem 0.1 we construct sequences of geometric objects; the nth term in the sequence of objects for any x ∈ W^{s}(p, 1) will lie in the nth neighborhood of p.

Let σ < 1 be a continuous function. For n ≥ 0 and p ∈ M, define the set S_{n,σ}(p) by:

S_{n,σ}(p) = ⋃_{x∈W^{s}(p,1)} Ŵ_{p}^{c}(x, σ_{n}(p)).

Lemma 4.2 Suppose that σ satisfies σ < min{γ̂, 1}. Then

f^{j}(S_{n,σ}(p)) ⊂ B(p_{j}, 2),

for j = 0, …, n.

Further, there exist positive constants κ < 1 and C > 0 such that, for every n ≥ 0,

f^{j}(S_{n,σ}(p)) ⊂ B(p_{j}, Cκ^{j}),

for j = 0, …, n.

Proof. Suppose that x ∈ W^{s}(p, 1) and y ∈ Ŵ_{p}^{c}(x, σ_{n}(p)). By part 3(a) of Proposition 3.1, we then have

y_{j} ∈ Ŵ_{p_{j}}^{c}(x_{j}, γ̂_{j}^{−1}(p)σ_{n}(p)) ⊂ Ŵ_{p_{j}}^{c}(x_{j}, 1) ⊂ B(p_{j}, 2),

for 0 ≤ j ≤ n. In fact, since σ < min{γ̂, 1}, the quantity γ̂_{j}^{−1}(p)σ_{n}(p) < γ̂_{j}^{−1}(p)σ_{j}(p) is exponentially small in j, as is the diameter of f^{j}(W^{s}(p, 1)). This implies the second conclusion. ⋄

Now let τ ≤ 1 be another continuous function. For every x ∈ S_{n,σ}(p), we have that B(x_{n}, τ_{n}(p)) ⊂ B(p_{n}, r), and so the set

T_{n,σ,τ}(p) = f^{−n} ( ⋃_{z∈f^{n}(S_{n,σ}(p))} Ŵ_{p_{n}}^{u}(z, τ_{n}(p)) ∪ W^{u}(z, τ_{n}(p)))

is well-defined. Proposition 3.1 and Lemma 1.1 imply that the leaves of Ŵ_{p_{j}}^{u} and W_{loc}^{u} are uniformly contracted by f^{−1} as long as they stay near the orbit of p; combining these facts with Lemma 4.2, we get:

Lemma 4.3 For every continuous function σ satisfying σ < min{γ̂, 1}, the set T_{n,σ,1}(p) satisfies

f^{j}(T_{n,σ,1}(p)) ⊂ B(p_{j}, 3),

for j = 0, …, n.

Further, for every such σ and every continuous function τ < 1, there exist positive constants κ < 1 and C > 0 such that, for every n ≥ 0,

f^{j}(T_{n,σ,τ}(p)) ⊂ B(p_{j}, Cκ^{j}),

for j = 0, …, n.

The dimensions of the neighborhoods T_{n,σ,τ}(p) and their iterates are illustrated in Figure 4.2, in the case where σ, τ < 1.

As simple corollary of this lemma and Lemma 4.1, we then obtain:

Lemma 4.4 Let α : M → R be a positive, uniformly Hölder continuous function, and let σ, τ be continuous functions satisfying

σ < min{γ̂, 1}, τ < 1.

Then there is a constant C ≥ 1 such that, for all n ≥ 0 and all x, y ∈ T_{n,σ,τ}(p),

C^{−1} ≤ \frac{α_{n}(y)}{α_{n}(x)} ≤ C.

PSfrag replacements T_{1,σ,τ}(p) T_{n,σ,τ}(p) T_{n+1,σ,τ}(p) f (T_{n,σ,τ}(p)) f (T_{1,σ,τ}(p)) f (T_{n+1,σ,τ}(p)) f^{n} (T_{n,σ,τ}(p)) f^{n} (T_{n+1,σ,τ}(p))

f^{n+1} (T_{n+1,σ,τ}(p))

p
Ŵ_{p}^{s}
Ŵ_{p}^{u}
Ŵ_{p}^{c}

Figure 1: Dimensions of the neighborhoods T_{n,σ,τ}(p) and their iterates

5 The main theorem

The properties of accessibility and essential accessibility can be reformulated using the notion of saturation. We say that a set is bisaturated if it is both W^{u}-saturated and W^{s}-saturated. Accessibility means that a set which is bisaturated must be either empty or all of M. Essential accessibility means that a measurable set which is bisaturated must have either 0 or full volume.

The central result of this paper is:

Theorem 5.1 Let f be C^{2}, partially hyperbolic and center bunched (or C^{1+δ} and strongly center bunched). Let A be a measurable set that is both essentially W^{u}-saturated and essentially W^{s}-saturated. Then the set of Lebesgue density points of A is W^{u}-saturated and W^{s}-saturated.

Theorem 5.1 does not assume that f is volume-preserving.

The central result of Pugh and Shub in [PS3] is a version of Theorem 5.1, which involves a different notion of density point (defined in [PS3]) and a slightly different hypothesis:

For a = u or s, if A is essentially W^{a}-saturated, then the set of julienne density points of A is W^{a}-saturated.

In contrast, Theorem 5.1 requires that A be both essentially W^{u}-saturated and essentially W^{s}-saturated in order to conclude anything.

With the obvious definitions, Theorem 5.1 has the corollary:

Corollary 5.2 If f is as in Theorem 5.1, then every bi essentially saturated set is essentially bisaturated.

Proof. Let A be a bi essentially saturated set. Theorem 5.1 implies that the set Â of Lebesgue density points of A is bisaturated. The Lebesgue Density Theorem implies that m(AΔÂ) = 0. Thus A is essentially bisaturated: it differs by a zero set from a bisaturated set.⋄

Essential accessibility tells us that essentially bi saturated sets have 0 or full measure. A priori, the notion of bi essential saturation is weaker than the notion of essential bisaturation. Corollary 5.2 says that the two concepts coincide when f is center bunched, so that if f is essentially accessible, then bi essentially saturated sets must have 0 or full measure.

Theorem 0.1 follows easily from Corollary 5.2 by a version of the Hopf argument and a result of Brin and Pesin (see Section 2 of [BPSW] for more details).

Proof of Theorem 0.1. Let μ be the f-invariant probability measure in the measure class of m. The concepts of μ-almost everywhere and m-almost everywhere are the same. To prove that f is ergodic with respect to μ, it suffices to show that the Birkhoff averages of continuous functions are almost everywhere constant. Let φ be a continuous function, and let

φ̂_{s}(p) = lim sup_{n→∞} \frac{1}{n} ∑_{i=0}^{n} φ(f^{i}(p)) and φ̂_{u}(p) = lim sup_{n→∞} \frac{1}{n} ∑_{i=0}^{n} φ(f^{−i}(p))

be the forward and backward Birkhoff averages of φ under f. The function φ̂_{s} is constant along W^{s}-leaves, and φ̂_{u} is constant along W^{u}-leaves. It follows that for any a ∈ R, the sets

A^{s}(a) = φ̂_{s}^{−1}(−∞, a] and A^{u}(a) = φ̂_{u}^{−1}(−∞, a].

are W^{s}-saturated and W^{u}-saturated, respectively.

Since f preserves μ, the Birkhoff Ergodic Theorem implies that φ̂_{s} = φ̂_{u} almost everywhere. Consequently m(A^{s}(a) Δ A^{u}(a)) = 0, so that the set A(a) = A^{u}(a) ∩ A^{s}(a) has A^{u}(a) as an essential W^{u}-saturate and A^{s}(a) as an essential W^{s}-saturate. Thus A(a) is bi essentially saturated.

It follows from Corollary 5.2 that A(a) is essentially bisaturated. Essential accessibility implies that A(a) has 0 or full measure. Since a was arbitrary, it follows that φ̂_{s} and φ̂_{u} are almost everywhere constant, and so f is ergodic.

To prove that f has the Kolmogorov property, it suffices to show that all sets in the Pinsker subalgebra P have 0 or full measure. According to Proposition 5.1 of [BP], if f is partially hyperbolic, then any set in P ∈ P is bi essentially saturated. It again follows from Corollary 5.2 and essential accessibility that P has 0 or full measure. ⋄

In order to prove Theorem 5.1 it suffices to show that the set of Lebesgue density points of A is W^{s}-saturated; applying this result with f replaced by f^{−1} then shows that the set of Lebesgue density points of A is also W^{u}- saturated. More precisely, it suffices to show that, for any p ∈ M, if x, x′ ∈ W^{s}(p, 1), and x is a Lebesgue density point of A, then so is x′.

Let

N = ⨆_{j≥0} B(p_{j}, r)

be the disjoint union, over j ≥ 0, of the balls B(p_{j}, r), where r ≫ 1 is given by Proposition 3.1. Everything we do in the rest of this paper takes place inside N, and we drop the dependence on p where it is not confusing.

We let Ŵ^{u} be the locally invariant foliation of N whose restriction to B(p_{j}, r) is Ŵ_{p_{j}}^{u}. Similarly we define foliations Ŵ^{c}, Ŵ^{s}, Ŵ^{cu} and Ŵ^{cs}. By Proposition 3.1, all of these foliations are uniformly Hölder continuous, Ŵ^{u} uniformly C^{1} subfoliates Ŵ^{cu}, and Ŵ^{s} uniformly C^{1} subfoliates Ŵ^{cs}. The foliations W^{u} and W^{s} induce foliations of N, which will again be denoted by W^{u} and W^{s}. Hence W^{s}(p) is used to denote the local leaf W^{s}(p, r). Note that Ŵ^{s}(p_{j}) = W^{s}(p_{j}) and Ŵ^{u}(p_{j}) = W^{u}(p_{j}) for all j ≥ 0. Note also that, since we are not assuming that f is dynamically coherent, there are no foliations W^{c}, W^{cu}, or W^{cs}.

Most leaves of these foliations (with the notable exception of the leaves of W^{s} and Ŵ^{s} passing through the orbit of p) are invariant under only finitely many iterates of f, until their orbits leave N. In our proof, the geometric objects in N that we need to iterate n times always start out in a neighborhood T_{n,σ,τ} of the type defined in Section 4.2 with τ < 1 and σ < min{γ̂, 1}. By Lemma 4.3, these objects remain inside of N for n iterates. As long as their orbits remain inside of N, the locally invariant foliations Ŵ^{u}, Ŵ^{c}, Ŵ^{s}, Ŵ^{cu} and Ŵ^{cs} used to construct these geometric objects are nearly indistinguishable from their invariant counterparts W^{u}, W^{c}, W^{s}, W^{cu} and W^{cs} in the dynamically coherent setting.

These locally invariant foliations differ from true invariant foliations in one key respect. The measurable set A in Theorem 5.1 is essentially W^{s}- saturated and essentially W^{u}-saturated. The property of (essential) W^{u}- saturation neither implies nor is implied by (essential) Ŵ^{u}-saturation. Similarly, (essential) W^{s}-saturation neither implies nor is implied by (essential) Ŵ^{s}-saturation. While it is possible to prove a version of Theorem 5.1 in which W^{u} and W^{s} are replaced by Ŵ^{u} and Ŵ^{s}, such a theorem is not enough to prove ergodicity using a Hopf argument. This is because the Hopf argument uses infinitely many iterates of f and is thus global in nature. Therefore, wherever we use explicitly the fact that our set A is W^{u}-saturated, in particular, in our arguments that use Proposition 2.5, we must switch from using Ŵ^{u} to using W^{u}, and similarly for stable foliations.

Following Pugh and Shub, we consider for each x ∈ W^{s}(p, 1) a sequence of sets, called center-unstable juliennes, that lie in the fake center-unstable manifold Ŵ^{cu}(x) and shrink exponentially as n → ∞ while becoming increasingly thin in the Ŵ^{u}-direction. In the Pugh-Shub construction, dynamical coherence is assumed, and the true invariant foliations W^{cu}, W^{c} and W^{u} are used; here we use their fake counterparts. While objects we work with, such as center-unstable juliennes, will depend on the fake invariant foliations given by Proposition 3.1, the final conclusion of Theorem 5.1 does not, since we always have Ŵ^{s}(p) = W^{s}(p).

Recall the center bunching assumptions (7):

ν < γγ̂ and ν̂ < γγ̂.

In what follows we will use only the first of these inequalities. The second inequality is used to prove W^{u} saturation of density points.

We choose continuous functions τ and σ such that

ν < τ < σγ and σ < min{γ̂, 1}.

Note that these inequalities also imply that

τν̂ < σγν̂ < σγγ̂ ≤ σ.

The choice of σ and τ with the desired properties is possible because of the center bunching assumption. The reader should think of τ as being just a little bigger than ν and σ as just a little bit less than min{γ̂, 1}. The reader might also choose to keep in mind the case where the functions ν, ν̂, γ, and γ̂ are constants, and where τ and σ can be chosen to be constant. In this case the cocycles τ_{n} and σ_{n} are just the constants τ^{n} and σ^{n}.

Using the notation defined in Section 4.2, let S_{n} = S_{n,σ}(p), and let T_{n} = T_{n,σ,τ}(p). For the rest of the paper, except where we indicate otherwise, cocycles will be evaluated at the point p. We will also drop the dependence on p from the notation; thus, if α is a cocycle, then α_{n}(p) will be abbreviated to α_{n}.

The center-unstable juliennes Ĵ_{n}^{cu}(x) that we construct will be contained in T_{n}. We now describe the construction of center-unstable juliennes.

Define, for all x ∈ W^{s}(p, 1),

B̂_{n}^{c}(x) = Ŵ^{c}(x, σ_{n}).

Note that

S_{n} = ⋃_{x∈W^{s}(p,1)} B̂_{n}^{c}(x).

For y ∈ S_{n}, we may then define two types of unstable juliennes:

Ĵ_{n}^{u}(y) = f^{−n}(Ŵ^{u}(y_{n}, τ_{n}))

and

J_{n}^{u}(y) = f^{−n}(W^{u}(y_{n}, τ_{n})).

Observe that for all y ∈ S_{n}, the sets Ĵ_{n}^{u}(y) and J_{n}^{u}(y) are contained in T_{n}.

For each x ∈ W^{s}(p, 1) and n ≥ 0, we then define the center-unstable julienne centered at x of order n:

Ĵ_{n}^{cu}(x) = ⋃_{y∈_{B_{n}^{c}}(x)} Ĵ_{n}^{u}(y).

Note that, by their construction, the sets Ĵ_{n}^{cu}(x) are contained in T_{n}, for all n ≥ 0 and x ∈ W^{s}(p, 1). Note also that Ĵ_{n}^{cu}(x) is contained in the smooth submanifold Ŵ^{cu}(x), which carries the restricted Riemannian volume m̂_{cu} = m_{W^{cu}}, and Ĵ_{n}^{cu}(x) has positive m̂_{cu}–measure. The reason for using Ĵ^{u} here instead of J^{u} is that the distribution T Ŵ^{c} ⊕ E^{u} is not necessarily integrable. The set defined by replacing Ĵ_{n}^{u} by J_{n}^{u} in the definition of Ĵ_{n}^{cu} might not be a C^{1} submanifold of M.

Our cu-juliennes are closely related to, but not exactly the same as, those of Pugh and Shub. In the case where σ and τ are constant functions, and f is dynamically coherent, their center-unstable julienne is the foliation product of W^{c}(x, σ^{n}) and f^{−n}(W^{u}(x_{n}, τ^{n})); see Figure 5. In this case, the image under f^{n} of our J_{n}^{cu}(p) appears in [PS3] as a tubelike approximation to the Pugh-Shub center-unstable postjulienne of rank n. The results of [PS3] show that the cu-juliennes defined here and in [PS3] are internested. Thus our cujuliennes could be replaced by the Pugh-Shub cu-juliennes in Propositions 5.3 and 5.5.

As in [PS3], the cu-juliennes have a quasi-conformality property: they are approximately preserved by holonomy along the stable foliation.

Proposition 5.3 Let x, x′ ∈ W^{s}(p, 1), and let h^{s} : Ŵ^{cu}(x) → Ŵ^{cu}(x′) be the holonomy map induced by the stable foliation W^{s}. Then the sequences h^{s}(Ĵ_{n}^{cu}(x)) and Ĵ_{n}^{cu}(x′) are internested.

There are estimates on the volumes of unstable and center-unstable juliennes.

PSfrag replacements B_{n}^{c} B̂_{n}^{c} J_{n}^{u} Ĵ_{n}^{u} x

Center-unstable julienne J_{n}^{cu}(x) in [PS3] W^{c} W^{u}

Ĵ_{n}^{cu}(x) in this paper.

Figure 2: Two types of center-unstable juliennes, when τ and σ are constant.

Proposition 5.4 There exist δ > 0 and c ≥ 1 such that, for all x ∈ W^{s}(p, 1), and all q, q′ ∈ S_{n}, the following hold, for all n ≥ 0:

c^{−1} ≤ \frac{m̂_{u}(Ĵ_{n}^{u}(q))}{m̂_{u}(Ĵ_{n}^{u}(q′))} ≤ c,

c^{−1} ≤ \frac{m_{u}(J_{n}^{u}(q))}{m_{u}(J_{n}^{u}(q′))} ≤ c,

m̂_{u}(Ĵ_{n+1}^{u}(q)) ≥ δm̂_{u}(Ĵ_{n}^{u}(q)),

and

m̂_{cu}(Ĵ_{n+1}^{cu}(x)) ≥ δm̂_{cu}(Ĵ_{n}^{cu}(x)).

The final crucial property of the cu-juliennes is that, for the sets that appear in the proof of Theorem 5.1, Lebesgue density points are precisely cu-julienne density points.

Proposition 5.5 Let X be a measurable set that is both W^{s}-saturated and essentially W^{u}-saturated. Then x ∈ W^{s}(p) is a Lebesgue density point of X if and only if:

lim_{n→∞} m̂_{cu}(X : Ĵ_{n}^{cu}(x)) = 1.

Note the asymmetry that X is fully W^{s}-saturated but only essentially W^{u}- saturated. This is because X will be an essential stable saturate of a bi essentially saturated set.

Remark: Pugh and Shub show that Lebesgue almost every point of any measurable set is a cu-julienne density point. In their argument they prove a Vitali covering lemma for their juliennes. This argument accounts for their definition of cu-juliennes as a foliation product and for the stronger bunching hypothesis in their main result. We do not know whether their result, specifically Theorem 7.1 of [PS3], still holds under our weaker bunching hypothesis, or whether it holds at all in the absence of dynamical coherence (although it can be shown that in the dynamically coherent, symmetrized setting that they consider, their hypothesis can be weakened from ν < γ^{2+2/θ_{0}} to ν < γ^{1/θ_{0}}).

The proof of Proposition 5.3 is essentially contained in [PS3]. For completeness we reproduce the argument in the next section. Proposition 5.4 is proved in Section 7, and the proof of Proposition 5.5 is in the final section. We now use these three propositions to prove the main result.

Proof of Theorem 5.1. As we noted above, it suffices to show that the Lebesgue density points of A are W^{s}-saturated; to see that the Lebesgue density points of A are W^{u}-saturated, just consider f^{−1} instead of f. Let A^{s} be an essential W^{s}-saturate of A. Since m(A Δ A^{s}) = 0, the Lebesgue density points of A are precisely the same as those of A^{s}. Fix p ∈ M and suppose that x ∈ W^{s}(p, 1) is a Lebesgue density point of A^{s}. Proposition 5.5 implies that x is a cu-julienne density point of A^{s}.

To finish the proof, we show that every x′ ∈ W^{s}(p, 1) is a cu-julienne density point of A^{s}. Then by Proposition 5.5, every x′ ∈ W^{s}(p, 1) is a Lebesgue density point of A^{s}. The Lebesgue density points of A^{s}, and hence of A, are therefore W^{s}-saturated.

Let h^{s} : Ŵ^{cu}(x) → Ŵ^{cu}(x′) be the holonomy map induced by the stable foliation W^{s}. The sequence h^{s}(Ĵ_{n}^{cu}(x)) ⊂ Ŵ^{cu}(x′) nests at x′. Transverse absolute continuity of h^{s} with bounded Jacobians implies that

lim_{n→∞} m̂_{cu}(A^{s} : Ĵ_{n}^{cu}(x)) = 1 ⟺ lim_{n→∞} m̂_{cu}(h^{s}(A^{s}) : h^{s}(Ĵ_{n}^{cu}(x))) = 1.

Since A^{s} is s-saturated, we then have:

lim_{n→∞} m̂_{cu}(A^{s} : Ĵ_{n}^{cu}(x)) = 1 ⟺ lim_{n→∞} m̂_{cu}(A^{s} : h^{s}(Ĵ_{n}^{cu}(x))) = 1.

Since we are assuming that x is a cu-julienne density point of A^{s}, we thus have

lim_{n→∞} m̂_{cu}(A^{s} : h^{s}(Ĵ_{n}^{cu}(x))) = 1.

Working inside of Ŵ^{cu}(x′), we will apply Lemma 2.1 to the sequences h^{s}(Ĵ_{n}^{cu}(x)) and Ĵ_{n}^{cu}(x′), which both nest at x′. Proposition 5.3 implies that these sequences are internested. Proposition 5.4 implies that Ĵ_{n}^{cu}(x′) is regular with respect to the induced Riemannian measure m̂_{cu} on Ŵ^{cu}(x′). Lemma 2.1 now tells us that

lim_{n→∞} m̂_{cu}(A^{s} : h^{s}(Ĵ_{n}^{cu}(x))) = 1 ⟺ lim_{n→∞} m̂_{cu}(A^{s} : Ĵ_{n}^{cu}(x′)) = 1,

and so x′ is a cu-julienne density point of A^{s}. It follows from Proposition 5.5 that x′ is a Lebesgue density point of A^{s}, and thus of A. ⋄

6 Julienne quasiconformality

We adapt the proof of Theorem 4.4 in [PS3] to prove Proposition 5.3. It will suffice to show that k can be chosen so that

h^{s}(Ĵ_{n}^{cu}(x)) ⊆ Ĵ_{n−k}^{cu}(x′),  (20)

for all n ≥ k, whenever x and x′ satisfy the hypotheses of the proposition. The hypotheses of the proposition treat x and x′ symmetrically, so we can then reverse their roles to obtain:

\overline{h}^{s}(Ĵ_{n}^{cu}(x′)) ⊆ Ĵ_{n−k}^{cu}(x),

for all n ≥ k, where \overline{h}^{s} : Ŵ_{loc}^{cu}(x′) → Ŵ^{cu}(x) is the holonomy induced also by the stable foliation W^{s}. Since \overline{h}^{s} and h^{s} are inverses, we then obtain:

Ĵ_{n}^{cu}(x′) ⊆ h^{s}(J_{n−k}^{cu}(x)),

for all n ≥ k.

In order to prove that k can be chosen so that (20) holds, we need two lemmas.

Lemma 6.1 There exists a positive integer k_{1} such that, for all x, x′ ∈ W^{s}(p),

ĥ^{s}(B̂_{n}^{c}(x)) ⊆ B̂_{n−k_{1}}^{c}(x′),

for all n ≥ k_{1}, where ĥ^{s} : Ŵ_{loc}^{c}(x) → Ŵ^{c}(x′) is the local Ŵ^{s} holonomy.

Proof. Proposition 3.1 implies that ĥ^{s} is L-Lipschitz, for some L ≥ 1. Therefore the image of Ŵ^{c}(x, σ_{n}) under ĥ^{s} is contained in Ŵ^{c}(x′, Lσ_{n}) ⊆ Ŵ^{c}(x′, σ_{n−k_{1}}), for any k_{1} large enough so that σ_{−k_{1}} > L. ⋄

Lemma 6.2 There exists a positive integer k_{2} such that the following holds for every integer n ≥ k_{2}. Suppose q ∈ S_{n}, q′ ∈ S_{n−k_{1}} and q′ ∈ Ŵ^{s}(q). Let y ∈ Ĵ_{n}^{u}(q), and let y′ be the image of y under W^{s} holonomy from Ŵ_{loc}^{cu}(q) to Ŵ^{cu}(q′). Then

y′ ∈ Ĵ_{n−k_{2}}^{u}(z′),

for some z′ ∈ Ŵ^{c}(q′, σ_{n−k_{2}}).

PSfrag replacements Ŵ^{u} Ŵ^{s} W^{s} Ŵ^{c}

p B̂_{n}^{c}(p)

q y q_{n} S_{n} y_{n} q′ y′ z′ q′_{n} y′_{n} z′_{n} f^{n}

O(τ_{n}) O(ν_{n})

Figure 3: Picture for the proof of Lemma 6.2.

Remark: Note that two types of holonomy maps appear in Lemma 6.2: the point q′ is the image of q under Ŵ^{s} holonomy between Ŵ_{loc}^{cu}(x) and Ŵ^{cu}(x′), whereas y′ is the image of y under W^{s} holonomy.

Proof of Lemma 6.2. Let z′ be the unique point in Ŵ^{u}(y′) ∩ Ŵ^{c}(q′). It is not hard to see that z′_{j} ∈ N, for j = 0, …, n − 1 and that z′_{n} is the unique point in Ŵ^{u}(y′_{n}) ∩ Ŵ^{c}(q′_{n}). It will suffice to prove that d(y′_{n}, z′_{n}) = O(τ_{n}) and d(q′, z′) = O(σ_{n}).

We have d(q_{n}, y_{n}) ≤ τ_{n} because y ∈ f^{−n}(Ŵ^{u}(q_{n}, τ_{n})). By Proposition 3.1, 3(a), we also have that d(q_{n}, q′_{n}) = O(ν_{n}) and d(y_{n}, y′_{n}) = O(ν_{n}), since d(q, q′) and d(y, y′) are both O(1). Note that q_{n} and z′_{n} are, respectively, the images of y_{n} and y′_{n} under Ŵ^{u}-holonomy between Ŵ_{loc}^{cs}(y_{n}) and Ŵ^{cs}(q_{n}). Uniform transversality of the foliations Ŵ^{u} and Ŵ^{cs} implies that

d(y′_{n}, z′_{n}) = O(max{d(q_{n}, y_{n}), d(y_{n}, y′_{n})}) = O(τ_{n}),

since ν < τ.

We next show that d(q′, z′) = O(σ_{n}). By the triangle inequality,

d(q′_{n}, z′_{n}) ≤ d(q′_{n}, q_{n}) + d(q_{n}, y_{n}) + d(y_{n}, y′_{n}) + d(y′_{n}, z′_{n}).

All four of the quantities on the right-hand side are easily seen to be O(τ_{n}). Since q′_{n} and z′_{n} lie in the same Ŵ^{c}-leaf at distance O(τ_{n}), Proposition 3.1 now implies that d(q′, z′) = O((γ_{n})^{−1}τ_{n}). But τ and σ were chosen so that τ < γσ. Hence (γ_{n})^{−1}τ_{n} < σ_{n} and d(q′, z′) = O(σ_{n}), as desired. ⋄

Proof of Proposition 5.3. As noted above, it suffices to prove the inclusion (20). For q ∈ B̂_{n}^{c}(x), let q′ = ĥ^{s}(q). Then q′ ∈ B̂_{n−k_{1}}^{c}(p′) by Lemma 6.1. Hence q, q′ ∈ S_{n−k_{1}} and we can apply Lemma 6.2 to obtain

h^{s}(Ĵ_{n}^{cu}(x)) ⊆ ⋃_{z∈Q} Ĵ_{n−k_{2}}^{u}(z),

where

Q = ⋃_{q′∈_{B_{n−k_{1}}^{c}}(x′)} B̂_{n−k_{2}}^{c}(q′).

For k ≥ k_{2}, we have:

⋃_{z∈Q} Ĵ_{n−k_{2}}^{u}(z) ⊆ ⋃_{z∈Q} Ĵ_{n−k}^{u}(z).

It therefore suffices to find k ≥ k_{2} such that Q ⊆ B̂_{n−k}^{c}(x′). This latter inclusion holds if:

σ_{n−k_{1}} + σ_{n−k_{2}} ≤ σ_{n−k},

which is obviously true for all n ≥ k, if k is sufficiently large. ⋄

7 Julienne measure

We next prove Proposition 5.4. Continuity of Ŵ^{u} implies that there exists C_{1} ≥ 1 such that

C_{1}^{−1} ≤ \frac{m̂_{u}(Ŵ^{u}(q_{n}, τ_{n}))}{m̂_{u}(Ŵ^{u}(q′_{n}, τ_{n}))} ≤ C_{1},  (21)

for all q, q′ ∈ S_{n}.

Let Ê^{s}, Ê^{c}, and Ê^{u} be the tangent distributions to the leaves of Ŵ^{s}, Ŵ^{c}, and Ŵ^{u}, respectively. They are Hölder continuous by Proposition 3.1, part 6. Furthermore, the restrictions of these distributions to T_{n} are invariant under T f^{j}, for j = 1, … n. We next observe that the Jacobian Jac(T f^{n}|_{E^{u}}) is nearly constant when restricted to the set T_{n}. More precisely, we have:

Lemma 7.1 There exists C_{2} ≥ 1 such that, for all n ≥ 1, and all y, y′ ∈ T_{n},

C_{2}^{−1} ≤ \frac{Jac(T f^{n}|_{E^{u}})(y)}{Jac(T f^{n}|_{E^{u}})(y′)} ≤ C_{2}.

Proof. By the Chain Rule, these inequalities follow from Lemma 4.4 with

α = Jac(T f|_{E^{u}}).⋄

Let q ∈ S_{n}, and let X ⊆ Ĵ_{n}^{u}(q) be a measurable set (such as Ĵ_{n}^{u}(q) itself). Then:

m̂_{u}(f^{n}(X)) = ∫_{X} Jac(T f^{n}|_{E^{u}})(x) dm̂_{u}(x).

From this and Lemma 7.1 we then obtain:

Lemma 7.2 There exists C_{3} > 0 such that, for all n ≥ 0, for any q, q′ ∈ S_{n}, and any measurable sets X ⊂ Ĵ_{n}^{u}(q), X′ ⊂ Ĵ_{n}^{u}(q′), we have:

C_{3}^{−1} \frac{m̂_{u}(f^{n}(X))}{m̂_{u}(f^{n}(X′))} ≤ \frac{m̂_{u}(X)}{m̂_{u}(X′)} ≤ C_{3} \frac{m̂_{u}(f^{n}(X))}{m̂_{u}(f^{n}(X′))}.

Recall that f^{n}(Ĵ_{n}^{u}(q)) = Ŵ^{u}(q_{n}, τ_{n}), for q ∈ S_{n}. The first conclusion of Proposition 5.4 now follows from (21) and Lemma 7.2 with X = Ĵ_{n}^{u}(q) and X′ = Ĵ_{n}^{u}(q′).

The second conclusion is proved similarly.

We next show that there exists δ > 0 such that

\frac{m̂_{u}(Ĵ_{n+1}^{u}(q))}{m̂_{u}(Ĵ_{n}^{u}(q))} ≥ δ,  (22)

for all n ≥ 0 and all q ∈ S_{n}. To obtain (22), we will apply Lemma 7.2 with q = q′, X = Ĵ_{n+1}^{u}(q), and X′ = Ĵ_{n}^{u}(q). This gives us:

\frac{m̂_{u}(Ĵ_{n+1}^{u}(q))}{m̂_{u}(Ĵ_{n}^{u}(q))} ≥ C_{3}^{−1} \frac{m̂_{u}(f^{n}(Ĵ_{n+1}^{u}(q)))}{m̂_{u}(f^{n}(Ĵ_{n}^{u}(q)))}.

But f^{n}(Ĵ_{n+1}^{u}(q)) = f^{−1}(Ŵ^{u}(q_{n+1}, τ_{n+1})) and f^{n}(Ĵ_{n}^{u}(q)) = Ŵ^{u}(q_{n}, τ_{n}), and hence:

\frac{m̂_{u}(f^{n}(Ĵ_{n+1}^{u}(q)))}{m̂_{u}(f^{n}(Ĵ_{n}^{u}(q)))} = \frac{m̂_{u}(f^{−1}(Ŵ^{u}(q_{n+1}, τ_{n+1})))}{m̂_{u}(Ŵ^{u}(q_{n}, τ_{n}))}.

This ratio is uniformly bounded below away from 0, since f^{−1} is a diffeomorphism, the leaves of Ŵ^{u} are uniformly smooth, and the ratio τ_{n+1}/τ_{n} = τ(p_{n}) is uniformly bounded below away from 0.

To prove the final claim, we begin by observing that, considered as a subset of Ŵ^{cu}(x), the set Ĵ_{n}^{cu}(x) fibers over B̂_{n}^{c}(x) with Ŵ^{u}-fibers Ĵ_{n}^{u}(q). We have just proved that these fibers are c-uniform. Since σ_{n+1}/σ_{n} = σ(p_{n}) is uniformly bounded away from 0, the ratio

\frac{m̂_{c}(B̂_{n+1}^{c}(x))}{m̂_{c}(B̂_{n}^{c}(x))} = \frac{Ŵ^{c}(x, σ_{n+1})}{Ŵ^{c}(x, σ_{n})}

is bounded away from 0, uniformly in x and n. Thus the sequence of bases B̂_{n}^{c}(x) of Ĵ_{n}^{cu}(x) is regular in the induced Riemannian volume m̂_{c}. Proposition 3.1, part 7. implies that Ŵ^{u} C^{1} subfoliates Ŵ^{cu}; in particular, considered as a subfoliation of Ŵ^{cu}(x), Ŵ^{u} is absolutely continuous with bounded Jacobians. Proposition 2.4 implies that the sequence Ĵ_{n}^{cu}(x) is regular, with respect to the induced Riemannian measure m̂_{cu}. This proves the final claim. ⋄

8 Julienne density

We now come to the proof of Proposition 5.5. We must show that if a measurable set X is both W^{s}-saturated and essentially W^{u}-saturated, then a point x ∈ W^{s}(p, 1) is a Lebesgue density point of X if and only if

lim_{n→∞} m̂_{cu}(X : Ĵ_{n}^{cu}(x)) = 1.

We will establish the following chain of equivalences:

x is a Lebesgue density point of X ⟺ lim_{n→∞} m(X : B_{n}(x)) = 1
⟺ lim_{n→∞} m(X : C_{n}(x)) = 1
⟺ lim_{n→∞} m(X : D_{n}(x)) = 1
⟺ lim_{n→∞} m(X : E_{n}(x)) = 1
⟺ lim_{n→∞} m(X : F_{n}(x)) = 1
⟺ lim_{n→∞} m(X : G_{n}(x)) = 1
⟺ lim_{n→∞} m̂_{cu}(X : Ĵ_{n}^{cu}(x)) = 1.

Before defining the sets B_{n}(x) through G_{n}(x), we outline the general scheme of the proof. After verifying the first equivalence, we prove that B_{n}(x) is regular, and that B_{n}(x) and C_{n}(x) are internested. The second equivalence then follows from Lemma 2.1. The sets C_{n}(x) and D_{n}(x) both fiber over the same base in Ŵ^{cs}, with c-uniform fibers in W^{u}, so the third equivalence follows from Proposition 2.5. We prove that the sets D_{n}(x), E_{n}(x), F_{n}(x), and G_{n}(x) are all internested, and that G_{n}(x) is a regular sequence. Equivalences 4-6 then follow from Lemma 2.1. Finally, G_{n}(x) fibers over Ĵ_{n}^{cu}(x), with cuniform W^{s}-fibers, and so the final equivalence follows from Proposition 2.7. This final step uses W^{s}-saturation of X.

The sets B_{n}(x) through G_{n}(x) are defined as follows. The set B_{n}(x) is a Riemannian ball in M:

B_{n}(x) = B(x, σ_{n}).

The sets C_{n}(x), D_{n}(x) and E_{n}(x) will fiber over the same base D_{n}^{cs}(x), where

D_{n}^{cs}(x) = ⋃_{x′∈_{W^{s}}(x,σ_{n})} B̂_{n}^{c}(x′).

Proposition 3.1, part 4. implies that D_{n}^{cs}(x) is contained in the C^{1} submanifold Ŵ^{cs}(x); the sequences D_{n}^{cs}(x) and Ŵ^{cs}(x, σ_{n}) are internested. Let

C_{n}(x) = ⋃_{q∈D_{n}^{cs}(x)} W^{u}(q, σ_{n}),

and let

D_{n}(x) = ⋃_{q∈D_{n}^{cs}(x)} J_{n}^{u}(q).

The set E_{n}(x) is nearly identical to D_{n}(x), with the crucial difference that the J_{n}^{u}-fibers are replaced with Ĵ_{n}^{u}-fibers:

E_{n}(x) = ⋃_{q∈D_{n}^{cs}(x)} Ĵ_{n}^{u}(q) = ⋃_{x′∈_{W^{s}}(x,σ_{n})} Ĵ_{n}^{cu}(x′) = ⋃_{x′∈W^{s}(x,σ_{n})} Ĵ_{n}^{cu}(x′).

The rightmost equality follows from the fact that Ŵ^{s}(x, σ_{n}) = W^{s}(x, σ_{n}), for all x ∈ W^{s}(p, 1) (Proposition 3.1, part 5.)

We define F_{n}(x) to be the foliation product of Ĵ_{n}^{cu}(x) and W^{s}(x, σ_{n}):

F_{n}(x) = ⋃_{q∈_{J_{n}^{cu}}(x), q′∈W^{s}(x,σ_{n})} W^{s}(q) ∩ Ŵ^{cu}(q′).

This definition makes sense since the foliations Ŵ^{cu} and W^{s} are transverse. Finally, let

G_{n}(x) = ⋃_{q∈_{J_{n}^{cu}}(x)} W^{s}(q, σ_{n}).

It is in the transition from D_{n} to E_{n} that the exchange between the measuretheoretically useful foliation W^{u} and the geometrically useful foliation Ŵ^{u} takes place. The definition of F_{n} is where we first use the foliation W^{s}.

Figure 8 is a schematic illustration of the relationship between the sets E_{n}(x), F_{n}(x) and G_{n}(x). All three sets contain Ĵ_{n}^{cu}(x) and W^{s}(x, σ_{n}). The set E_{n}(x) fibers over W^{s}(x, σ_{n}) with fibers of the form Ĵ_{n}^{cu}(·). The set G_{n}(x) fibers over Ĵ_{n}^{cu}(x) with fibers of the form W^{s}(·, σ_{n}). The foliation product F_{n}(x) of Ĵ_{n}^{cu}(x) and W^{s}(x, σ_{n}) is, in some sense, intermediate between E_{n}(x) and G_{n}(x).

We now prove these equivalences, following the outline described above. First, recall that B_{n}(x) is a round ball about x of radius σ_{n}. The forward

implication in the first equivalence is obvious from the definition of B_{n}(x). The backward implication follows from this definition and the fact that the ratio σ_{n+1}/σ_{n} = σ(p_{n}) of successive radii is less than 1, and is bounded away from both 0 and 1 independently of n. From this we also see that B_{n}(x) is regular.

The set C_{n}(x) fibers over D_{n}^{cs}(x), with fiber W^{u}(x′, σ_{n}) over x′ ∈ D_{n}^{cs}(x). The sequence D_{n}^{cs}(x) internests with the sequence of disks Ŵ^{cs}(x, σ_{n}), by continuity and transversality of the foliations Ŵ^{c} and Ŵ^{s}. Continuity and transversality of the foliations W^{u} and Ŵ^{cs} then imply that C_{n}(x) and B_{n}(x) are internested.

PSfrag replacements

W^{s}
Ŵ^{cu}
x
E_{n}(x)
F_{n}(x)
G_{n}(x)
Ĵ_{n}^{cu}(x)
Ŵ^{s}(x, σ_{n})

Figure 4: Comparison between E_{n}(x), F_{n}(x) and G_{n}(x).

To prove the equivalence

lim_{n→∞} m(X : C_{n}(x)) = 1 ⟺ lim_{n→∞} m(X : D_{n}(x)) = 1,

we note that C_{n}(x) and D_{n}(x) both fiber over D_{n}^{cs}(x), with W^{u}-fibers. Since X is essentially W^{u}-saturated, Proposition 2.5 implies that it suffices to show that the fibers of C_{n}(x) and D_{n}(x) are both c-uniform. The fibers of of C_{n}(x) are easily seen to be uniform, because they are all comparable to balls in W^{u} of fixed radius σ_{n}. The fibers of D_{n}(x) are the unstable juliennes J_{n}^{u}(x′), for x′ ∈ D_{n}^{cs}(x). Uniformity of these fibers follows from Proposition 5.4.

We next prove:

Lemma 8.1 The sequences D_{n}(x) and E_{n}(x) are internested.

Proof. Recall that

D_{n}(x) = ⋃_{q∈D_{n}^{cs}(x)} J_{n}^{u}(x), and E_{n}(x) = ⋃_{q∈D_{n}^{cs}(x)} Ĵ_{n}^{u}(x).

Internesting of the sequences D_{n}(x) and E_{n}(x) means that there is a k ≥ 0 such that, for all n ≥ k,

D_{n}(x) ⊆ E_{n−k}(x) and E_{n}(x) ⊆ D_{n−k}(x).

We will show that there is a k for which the first inclusion holds. Reversing the roles of of W^{u} and Ŵ^{u} in the proof gives the second inclusion.

Suppose y ∈ D_{n}(x). Then y ∈ J_{n}^{u}(q) = f^{−n}(W^{u}(q_{n}, τ_{n})), for some q ∈ D_{n}^{cs}(x); in particular,

d(y_{n}, q_{n}) = O(τ_{n}).  (23)

Let q̂ be the unique point of intersection of Ŵ^{u}(y) with Ŵ^{cs}(x). We will show that y ∈ E_{n−k}(x), for some k that is independent of n. In order to do this, it suffices to show that q̂ ∈ D_{n−k}^{cs}(x) and y ∈ Ĵ_{n−k}^{u}(q̂) = f^{−(n−k)}(Ŵ^{u}(q̂_{n−k}, τ_{n−k})).

In order to prove that q̂ ∈ D_{n−k}^{cs}(x) it will suffice to show that

d(q, q̂) = o(σ_{n})  (24)

(in fact, O(σ_{n}) would suffice, but the argument gives o(σ_{n})). In order to prove that y ∈ Ĵ_{n−k}^{u}(q̂) it will suffice to show that

d(y_{n}, q̂_{n}) = O(τ_{n}).  (25)

Equation (24) follows easily from (25). Since y_{n} and q̂_{n} lie in the same Ŵ^{u} leaf, Proposition 3.1 and (25) imply that

d(y, q̂) = O(ν̂_{n}τ_{n}) = o(σ_{n}),  (26)

since ν̂τ < σ. Similarly, Proposition 3.1 and (23) imply that

d(y, q) = o(σ_{n}).  (27)

Applying the triangle inequality to (26) and (27) gives (24).

It remains to prove (25). Recall from the construction of the fake foliations in Proposition 3.1 that, at any point z in the neighborhood N of the orbit of p in which the fake foliations are defined, the tangent space T_{z}Ŵ^{u}(z) lies in the ε-cone about T_{z}W^{u}(z) = E^{u}(z). Furthermore, the angle between T_{z}Ŵ^{cs}(z) and either T_{z}Ŵ^{u}(z) or T_{z}W^{u}(z) is uniformly bounded away from 0. Note that q̂_{n} is the unique point in Ŵ^{u}(y_{n})∩Ŵ^{cs}(x_{n}) and q_{n} is the unique point in W^{u}(y_{n}) ∩ Ŵ^{cs}(x_{n}); combining this with (23) gives:

d(y_{n}, q̂_{n}) = O(d(y_{n}, q_{n})) = O(τ_{n}).

This completes the proof.⋄

We next show:

Lemma 8.2 E_{n}(x) and F_{n}(x) are internested, as are F_{n}(x) and G_{n}(x).

Proof. The sets E_{n}(x) and F_{n}(x) both fiber over the same base Ŵ^{s}(x, σ_{n}). The fibers of E_{n}(x) are the cu-juliennesĴ_{n}^{cu}(x′), for x′ ∈ Ŵ^{s}(x, σ_{n}). the fibers of F_{n}(x) are images of Ĵ_{n}^{cu}(x) under W^{s}-holonomy from Ŵ^{cu}(x) to Ŵ^{cu}(x′), for x′ ∈ Ŵ^{s}(x, σ_{n}). It follows immediately from Proposition 5.3 that the sequences E_{n}(x) and F_{n}(x) are internested.

To see that F_{n}(x) and G_{n}(x) are internested, suppose that q′ lies in the boundary of the fiber of F_{n}(x) that lies in W^{s}(q) for some q ∈ Ĵ_{n}^{cu}(x). Then q′ ∈ Ĵ_{n}^{cu}(x′) for a point x′ that lies in the boundary of W^{s}(x, σ_{n}). The diameters of Ĵ_{n}^{cu}(x) and Ĵ_{n}^{cu}(x′) are both O(σ_{n}), and d(x, x′) = σ_{n}. Hence, if k is large enough, we will have

σ_{n+k} ≤ d(q, q′) ≤ σ_{n−k}.

Thus all points on the boundary of the fiber of F_{n}(x) in W_{loc}^{s}(q) lie outside W^{s}(q, σ_{n+k}) and inside W^{s}(q, σ_{n−k}). ⋄

We now know that any two of D_{n}(x), E_{n}(x), F_{n}(x) and G_{n}(x) are internested. As discussed above, to prove the fourth through sixth equivalences, it now suffices to show:

Lemma 8.3 The sequence G_{n}(x) is regular for each x ∈ W^{s}(p, 1).

Proof. The set

G_{n}(x) = ⋃_{q∈_{J_{n}^{cu}}(x)} W^{s}(q, σ_{n})

fibers over Ĵ_{n}^{cu}(x), with W^{s}-fibers W^{s}(q, σ_{n}). Since W^{s} is absolutely continuous, Proposition 2.4 implies that regularity of G_{n}(x) follows from regularity of the base sequence and fiber sequence. Proposition 5.4 implies that the sequence Ĵ_{n}^{cu}(x) is regular in the induced measure m̂_{cu}. As we remarked above, the ratio σ_{n+1}/σ_{n} = σ(p_{n}) is uniformly bounded below away from 0. Consequently, the ratio

\frac{m_{s}(W^{s}(q, σ_{n+1}))}{m_{s}(W^{s}(q, σ_{n}))}

is bounded away 0, uniformly in x, q, and n. The regularity of G_{n}(x) now follows from Proposition 2.4. ⋄

To prove the final equivalence, we use the fact that G_{n}(x) fibers over Ĵ_{n}^{cu}(x) with c-uniform fibers and apply Proposition 2.7. Here we use the fact that X is W^{s}-saturated. This completes the proof of Proposition 5.5. ⋄

9 Open questions

We do not know whether the bunching assumption can be dropped in Theorem 0.1. It seems clear that a significant new idea would be required to remove this hypothesis. Indeed, even if we impose stringent assumptions on the regularity of the center, center-stable and center-unstable foliations, as Brin and Pesin do in their 1974 work, every known ergodicity result requires the center bunching hypothesis (7). Thus a first step in answering this question might be to answer the following question:

Question: Suppose that f is C^{2}, volume-preserving and partially hyperbolic. If E^{c} is absolutely continuous (Lipschitz, smooth, …) and f is (essentially) accessibile, is f then ergodic?

As a concrete example, consider a diffeomorphism f_{λ} : T^{2}×T^{2} → T^{2}×T^{2} of the form:

f_{λ}(x, y) = (A(x), g_{λ}(y)),

where A : T^{2} → T^{2} is the linear Anosov diffeomorphism given by

A = ( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} )^{2} ,

and g_{λ} : T^{2} → T^{2} is a standard map of the form:

g_{λ}(z, w) = (z + w, w + \frac{λ}{2π} sin(2π(z + w))).

A straightforward calculation shows that there is an interval Λ ⊂ R containing (−4, 4) such that, if λ ∈ Λ, then f_{λ} is partially hyperbolic with respect to the standard (flat) metric on T^{2} × T^{2}. It is also not difficult to show, by examining the spectrum of T f_{λ} at the fixed point (0, 0), that f_{λ} is center bunched if and only if λ ∈ (−1, 1).

This example appears in [SW1], where it is shown that there is a function φ : T^{2} → T^{2} with φ(0) = 0 and an interval E = (0, ϵ_{0}) such that, for all (λ, ϵ) ∈ Λ × E, the map

f_{λ,ϵ}(x, y) = (A(x), g_{λ}(y) + ϵφ(x))

is both partially hyperbolic and stably accessible. Furthermore, f_{λ,ϵ} is center bunched if and only if (λ, ϵ) ∈ (−1, 1) × E. For all (λ, ϵ) ∈ Λ × E, the center bundle E^{c} is tangent to the fibers {x} × T^{2} and is C^{∞}. The foliations W^{cu} and W^{cs} are also C^{∞}.

Theorem 0.1 implies that f_{λ,ϵ} is stably ergodic for all (λ, ϵ) ∈ (−1, 1)×E. We do not however know whether f_{λ,ϵ} is ergodic for even a single value of (λ, ϵ) ∈ (Λ \ (−1, 1)) × E.

We also do not know whether or not the center bunching inequalities (7) imply that a partially hyperbolic diffeomorphism is dynamically coherent. The simplest non dynamically coherent examples are linear Anosov diffeomorphisms of nilmanifolds, viewed as partially hyperbolic by grouping the weaker stable and unstable directions together to form a center distribution. It was first observed in [W] that this construction can provide examples that are not dynamically coherent.

In [BW3], we discuss a family of examples of this type. They are Anosov diffeomorphisms of a compact quotient of the product of the Heisenberg group with itself induced by linear maps whose eigenvalues are

λ^{a+b} ≥ λ^{b} ≥ λ^{a} ≥ λ^{−a} ≥ λ^{−b} ≥ λ^{−a−b},

where b ≥ a ≥ 0 and λ > 1 is the larger of the two eigenvalues of a hyperbolic matrix in SL(2, Z) whose eigenvalues are λ > 1 > λ^{−1} > 0. When a > 0, the map can be viewed as a partially hyperbolic diffeomorphism with E^{u} and E^{s} being the left invariant one dimensional distributions corresponding to the eigenvalues λ^{a+b} and λ^{−a−b} respectively. The center is the left invariant four dimensional distribution corresponding to the sum of the eigenspaces for the other eigenvalues.

The linear map is chosen so that [E^{c}, E^{c}] is spanned by E^{u} ⊕ E^{s}. This ensures that distribution E^{c} is not integrable. Consequently these examples are not dynamically coherent.

For these examples we have

ν = ν̂ = λ^{−a−b} and γ = γ̂ = λ^{−b}.

Thus center bunching fails, because

ν = ν̂ = λ^{−a−b} ≥ λ^{−2b} = γγ̂.

Note that center bunching only just fails in the extreme case when a = b. Gourmelon has constructed perturbations of the a = b example that are robustly ergodic, but neither Anosov nor dynamically coherent. It does not seem to be known whether or not his perturbations are center bunched.

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