A note on stable holonomy between centers

Preliminary version

Keith Burns and Amie Wilkinson

June 11, 2005

There is a slight disparity in smooth ergodic theory, between Pesin theory and the Pugh-Shub partially hyperbolic theory. Both theories assume a weakened form of hyperbolicity and conclude ergodicity. While the two theories have considerable overlap, Pesin theory assumes a C^{1+δ} hypothesis, whereas the Pugh-Shub stable ergodicity theorem assumes a C^{2} hypothesis. The purpose of this paper is to close the gap between C^{1+δ} and C^{2}.

Let f : M → M be a partially hyperbolic diffeomorphism of a compact manifold M. Partially hyperbolic means the following. There is a nontrivial splitting of the tangent bundle, T M = E^{u} ⊕ E^{c} ⊕ E^{s}, 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, and γ̂ with

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

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

∥T fv∥ ≤ ν(p),  if v ∈ E^{s}(p),  (2)
γ(p) ≤ ∥T fv∥ ≤ γ̂(p)^{−1},  if v ∈ E^{c}(p),  (3)
ν̂(p)^{−1} ≤ ∥T fv∥,  if v ∈ E^{u}(p).  (4)

A partially hyperbolic diffeomorphism is dynamically coherent if there are foliations W^{cs} and W^{cu} tangent to E^{c} ⊕ E^{s} and E^{c} ⊕ E^{u} respectively. In this case there is also a foliation W^{c} tangent to E^{c} whose leaves are the intersections of the leaves of W^{cs} and W^{cu}. Each leaf of W^{cs} is foliated by leaves of W^{c} and W^{s}; leaves of W^{cu} have the analogous property.

Theorem 0.1 ([PSW], Theorem B) Let f : M → M be a C^{2}, partially hyperbolic, dynamically coherent diffeomorphism of a compact manifold M satisfying

ν < γγ̂.

Then the stable distribution is a uniformly C^{1} bundle when restricted to any leaf of the center-stable foliation. Consequently, the stable holonomy map between center transversals is C^{1}.

If the assumption that f is C^{2} is relaxed, then the first conclusion of Theorem 0.1 does not hold; that is, the stable bundle may no longer be a C^{1} bundle over every leaf of the center-stable foliation. This is easy to see in the case where the unstable bundle is trivial. Let F : T^{2} → T^{2} be a linear Anosov diffeomorphism of the 2-torus, and let f = φ ∘ F ∘ φ^{−1}, where φ : T^{2} → T^{2} is a C^{1+δ} diffeomorphism that is not C^{2}. Then f is an Anosov diffeomorphism, with E_{f}^{s} = φ_{∗}(E_{F}^{s}) and E_{f}^{u} = φ_{∗}(E_{F}^{u}). We regard f as a partially hyperbolic diffeomorphism, with stable bundle E_{f}^{s} and center bundle E_{f}^{u}. Then the center-stable foliation has one leaf, the whole 2-torus, and the stable bundle E_{f}^{s} = φ_{∗}(E_{F}^{s}) is merely Hölder continuous, since T φ is Hölder-continuous, but not C^{1}. Note that the hypothesis ν < γγ̂ is satisfied by this example, since the center bundle of f is 1-dimensional.

While the first conclusion of Theorem 0.1 does not hold for this example, the second one does; the stable holonomy for f between center transversals is C^{1}, and in fact, C^{1+δ}. This is because the stable holonomy for f is conjugate via φ to the stable holonomy for F.

Note also that, even under the hypothesis that f is C^{2}, the techniques behind the proof of Theorem 0.1 do not give a C^{1+β} condition on the stable holonomy; for this, a C^{2+δ} hypothesis on f is required. Our main result shows that we may relax the C^{2} assumption to C^{1+δ} and still obtain that the stable holonomy for f between center transversals is C^{1}. The price of relaxing the differentiability of f is that we require a more stringent bunching condition, a condition that is nonetheless satisfied by many examples. As an added benefit, we obtain that the holonomy is C^{1+β}, for some β > 0.

Theorem 0.2 Let f : M → M be a C^{1+δ}, partially hyperbolic, dynamically coherent diffeomorphism of a compact manifold M satisfying

ν^{θ} < γγ̂,

where θ ∈ (0, δ) is a Hölder exponent for E^{c}. Then there exists β > 0 such that the stable holonomy map between center transversals is C^{1+β}.

As far as we know, this result is new, although there are related results for codimension-1 hyperbolic systems [PR], in the nonuniform setting when E^{c} is a hyperbolic Lyapunov subspace in an Oseledec decomposition [BPS], and for Lie group cocycles [PW] .

For any partially hyperbolic diffeomorphism, the stable and unstable subbundles E^{s} and E^{u} are tangent to foliations, which we denote by W^{s} and W^{u} respectively [BP]. These foliations induce an equivalence relation on M: we say that p ∼_{us} q if there is a sequence of points p = p_{0}, …, p_{k} = q such that any two consecutive points in the sequence lie in the same W^{s}-leaf or the same W^{u}-leaf. A partially hyperbolic diffeomorphism has the accessibility property if there is only one ∼_{us}-class, i.e. if p ∼_{us} q for any p, q ∈ M. It has the essential accessibility property if a set that is measurable (with respect to the volume) and is a union of ∼_{us}-classes must have 0 or full measure. In [BW], we use Theorem 0.2 to prove the following:

Theorem 0.3 Let f be C^{1+δ}, volume preserving, and partially hyperbolic. Suppose that f satisfies the strong center bunching condition:

max{ν, ν̂}^{θ} < γγ̂,  (5)

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

νγ^{−1} < μ^{θ},  ν̂γ̂^{−1} < μ̂^{θ}.  (6)

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

This has the corollary:

Corollary 0.4 Let f : M → M be a C^{1+δ}, volume-preserving, partially hyperbolic, dynamically coherent diffeomorphism with 1-dimensional center bundle. If f is accessible, then f is ergodic.

Note that Theorem 0.2 can equally be applied to the stable foliation of C^{1+δ} Anosov diffeomorphism by regarding the unstable bundle as a center bundle. By doing so, we obtain, for example, the following:

Corollary 0.5 Let f be a C^{1+δ} conformal Anosov diffeomorphism. Then for every β < δ, there exists a neighborhood U of f in Diff^{1+δ}(M), such that, for every g ∈ U, the stable and unstable holonomies for g are C^{1+β}.

1 Proof of Theorem 0.2

The proof follows similar lines to the proof of absolute continuity of the stable foliation for partially hyperbolic systems. The main difference is that, unlike the Jacobian map x ↦ Jac_{x}f, the multiplicative cocycle x ↦ T_{x}f over f is not abelian. We exploit instead the fact that the map v ↦ ∥T fv∥ is an abelian multiplicative cocycle over the projectivized tangent map f_{∗} : P (T M) → P (T M). This map is projective in the fibers, with Lipschitz norm (γγ̂)^{−1}. The inequality ν^{θ} < γγ̂ allows us to prove that the stable holonomy h^{s} has a projectivized derivative h_{∗}^{s}. Once this is established, a distortion estimate using the v ↦ ∥T fv∥ cocycle gives us that h^{s} is differentiable.

We denote by h^{s} the stable holonomy between two center leaves, omitting reference to domain and codomain. We denote by π^{s} a fixed, uniformly C^{1+θ}, local approximation to h^{s}. By this we mean that there is a constant C > 0 such that for any p, q ∈ M with q ∈ W^{s}(p, 1) we have a C^{1+θ} diffeomorphism

π_{p,q}^{s} : W_{loc}^{c}(p) → W^{c}(q)

such that:

1. d(π_{p,q}^{s}(p), q) ≤ Cd(p, q);

2. d(T π_{p,q}^{s}v, v) ≤ Cd(p, q)^{θ}, for all v ∈ P (E^{c}(p));

3. if W_{loc}^{c}(p) ∩ W_{loc}^{c}(p′) ≠ ∅, and the codomains of π_{p,q}^{s} and π_{p′,q′}^{s} also intersect nontrivially, then π_{p,q}^{s} = π_{p′,q′}^{s} on W_{loc}^{c}(p) ∩ W_{loc}^{c}(p′).

This can be accomplished by fixing a C^{1+θ} normal bundle to each leaf of the center foliation W^{c}. Since the leaves of W^{c} vary continuously in the C^{1+θ} topology, this choice of normal bundle can be made to vary continuously from leaf to leaf. The fibers of this bundle foliate a uniform neighborhood of each leaf. The holonomy of this local foliation between a local center leaf W_{loc}^{c}(p) and a neighboring center leaf W_{loc}^{c}(q) gives the map π_{p,q}^{s}. By rescaling the metric if necessary, we may assume that π_{p,q}^{s} is well-defined on the domain W_{loc}^{c}(p), for all q ∈ W^{s}(p, 1). We will sometimes drop the reference to p and q and write π^{s} for short.

Fix p, q ∈ M with q ∈ W^{s}(p, 1), and let

h_{n}^{s} = f^{−n} ∘ π_{p_{n},q_{n}}^{s} ∘ f^{n}.

Note that h_{n}^{s} : W_{loc}^{c}(p) → W^{c}(q) is well-defined, for all n ≥ 0, since π_{p_{n},q_{n}}^{s} extends uniquely to the domain f^{n}(W_{loc}^{c}(p)).

The proof now proceeds in four steps alluded to above. In the first step, we show that h_{n}^{s} converges in the C^{0} topology to h^{s}. This first step uses only partial hyperbolicity and dynamical coherence. In the second step, we show that the projective linear bundle maps h_{n}^{s}_{∗} : P (E^{c}) → P (E^{c}) converge in the C^{0} topology to a projective linear bundle map h_{∗}^{s} : P (E^{c}) → P (E^{c}). In Step 3, we show that the derivatives T h_{n}^{s} : E^{c} → E^{c} converge in the C^{0} topology to a bundle isomorphism T h^{s} : E^{c} → E^{c}. It follows that h^{s} is C^{1}, with derivative T h^{s}. Finally, we show that T h^{s} is Hölder continuous. The last three steps use the bunching hypothesis ν^{θ} < γγ̂.

Step 1: the sequence h_{n}^{s} converges to h^{s} in the space of all continuous maps from W_{loc}^{c}(p) to W^{c}(q).

Invariance of W^{s} under f implies that f^{−n}h_{p_{n},q_{n}}^{s}f^{n} = h^{s}, where the stable holonomy h_{p_{n},q_{n}}^{s} maps from f^{n}(W_{loc}^{c}(p)) to W^{c}(q_{n}), sending p_{n} to q_{n}.

Note that since f^{n} contracts distances in W^{s}(x, 1) by a factor of ν_{n}(x), we have that d(x_{n}, h_{p_{n},q_{n}}^{s}(x_{n})) ≤ ν_{n}(x), for all x ∈ W_{loc}^{c}(p) and n ≥ 1. By property 1. of π^{s}, we then obtain that

d(π^{s}(x_{n}), h_{p_{n},q_{n}}^{s}(x_{n})) ≤ Cν_{n}(x),  (7)

for all x ∈ W_{loc}^{c}(p) and n ≥ 1.

Now fix x ∈ W_{loc}^{c}(p) and n ≥ 1. We have:

d(h_{n}^{s}(x), h^{s}(x)) = d(f^{−n}π^{s}(x_{n}), f^{−n}h_{p_{n},q_{n}}^{s}(x_{n}))
≤ γ_{−n}(x) d(π^{s}(x_{n}), h_{p_{n},q_{n}}^{s}(x_{n}))
≤ Cγ_{−n}(x)ν_{−n}(x).

Partial hyperbolicity implies that there exists κ < 1 such that the function γ_{−n}ν_{n} is uniformly bounded by κ^{n}. Hence h_{n}^{s} converges uniformly to h^{s}.⋄

Step 2: the sequence h_{n}^{s}_{∗} : P (T W_{loc}^{c}(p)) → P (T W^{c}(q)) is Cauchy.

We fix some notation. For w ∈ P (T M) and k ∈ Z, let w_{k} = f_{∗}^{k}(w) ∈ P (T M). We first prove some lemmas.

Lemma 1.1 There exists δ > 0 such that, for every v ∈ P (E^{c}(x)) and w ∈ P (E^{c}(y)):

1. if d(v_{k}, w_{k}) ≤ δ, for k = 0, …, n − 1, then

d(v_{n}, w_{n}) ≤ (γγ̂)_{−n}(x_{n})d(v, w)^{θ};

2. if d(v_{−k}, w_{−k}) ≤ δ, for k = 0, …, n − 1, then

d(v_{−n}, w_{−n}) ≤ (γγ̂)_{−n}(x)d(v, w)^{θ}.

In particular, if d(v, w) ≤ δ(γγ̂)_{n}(x), then the conclusions in 1. hold, and if d(v, w) ≤ δ(γγ̂)_{n}(x_{n}), then the conclusions in 2. hold.

Proof. It suffices to prove the lemma in the case n = 1; the other cases are proved inductively. In writing f_{∗}(v) for v ∈ S(T_{x}M), we deliberately suppress the basepoint x. For the following calculation, the basepoint is relevant, so we shall write v = (x, ζ) and w = (y, ξ). Recall that the Lipschitz norm of f_{∗} restricted to the fiber P (E^{c}(x)) is γγ̂(x)^{−1} = (γγ̂)_{−1}(x_{1}), and the Lipschitz norm of f_{∗}^{−1} restricted to the fiber P (E^{c}(x)) is γγ̂(x_{−1})^{−1} = (γγ̂)_{−1}(x). We estimate:

d(v_{1}, w_{1}) = d(f_{∗}(x, ζ), f_{∗}(y, ξ))
≤ d(f_{∗}(x, ζ), f_{∗}(x, ξ)) + d(f_{∗}(x, ξ), f_{∗}(y, ξ))
≤ (γγ̂)_{−1}(x_{1})d(ζ, ξ) + Hd(x, y)^{δ}
≤ (γγ̂)_{−1}(x_{1}) (d(ζ, ξ)^{1−θ} + Hd(x, y)^{δ−θ}) sup{d(x, y), d(ζ, ξ)}^{θ}.

If d(v, w) = sup{d(ζ, ξ), d(x, y)} is sufficiently small, then

d(ζ, ξ)^{1−θ} + Hd(x, y)^{δ−θ} < 1,

and we have

d(v_{1}, w_{1}) ≤ (γγ̂)_{−1}(x_{1})d(v, w)

This completes the proof of 1, in the case n = 1. The case 2. is proved similarly.⋄

The next Lemma will also be useful in Step 3.

Lemma 1.2 For n ≥ 1, and k ≤ n, let

v′_{k} = f_{∗}^{−n+k}π_{∗}(v_{n}),

where v ∈ P (T_{x}W_{loc}^{c}(p)). Then, for n sufficiently large, we have:

d(v_{k}, v′_{k}) ≤ Cν_{k}(x)^{θ},

for all k ≤ n.

Proof. Property 2. of π^{s} and uniform contraction of W^{s}(p, 1) by ν together imply that

d(v_{n}, v′_{n}) = d(v_{n}, π_{∗}^{s}(v_{n}))
≤ Cd(x_{n}, h_{p_{n},q_{n}}^{s}(x_{n}))^{θ}
≤ Cν_{n}(x)^{θ}.

Let δ > 0 be given by Lemma 1.1. The bunching hypothesis ν^{θ} < γγ̂ implies that if n is sufficiently large, we have:

d(v_{n}, v′_{n}) ≤ δ(γγ̂)_{n}(x).

Assume n is this large.

We prove Lemma 1.2 by backward induction on k. We have shown that

d(v_{n}, v′_{n}) ≤ min{Cν_{n}(x)^{θ}, δ(γγ̂)_{n}(x)}.

Lemma 1.1 now implies that for all 1 ≤ k ≤ n, we have

d(v_{k}, v′_{k}) = d(f_{∗}^{−n+k}(v_{n}), f_{∗}^{−n+k}(v′_{n}))
≤ (γγ̂)_{k−n}(x)d(v_{n}, v′_{n})
≤ δ(γγ̂)_{k}(x)

Suppose now that for some k we have

d(v_{k}, v′_{k}) ≤ Cν_{k}(x)^{θ}.

Then, since d(v_{k}, v′_{k}) ≤ δ, we have

d(v_{k−1}, v′_{k−1}) = d(f_{∗}^{−1}(v_{k}), f_{∗}^{−1}(v′_{k}))
≤ (γγ̂)_{−1}(x_{k})d(v_{k}, v′_{k})
≤ (γγ̂)_{−1}(x_{k})Cν_{k}(x)^{θ}
≤ Cν_{k−1}(x)^{θ},

since ν^{θ} < γγ̂. ⋄

Returning to the proof of Step 2, we fix v ∈ P (T W_{loc}^{c}(p)), and n ≥ 1. Then

d(h_{n}_{∗}(v), h_{n+1}_{∗}(v)) = d(f_{∗}^{−n}π_{∗}^{s}(v_{n}), f_{∗}^{−n−1}π_{∗}^{s}f_{∗}(v_{n}))
≤ (γγ̂)_{−n}(x) d(π_{∗}^{s}(v_{n}), f_{∗}^{−1}π_{∗}^{s}f_{∗}(v_{n}))
≤ (γγ̂)_{−n}(x) [d(π_{∗}^{s}(v_{n}), (v_{n})) + d(f_{∗}^{−1}f_{∗}(v_{n}), f_{∗}^{−1}π_{∗}^{s}f_{∗}(v_{n})]
≤ (γγ̂)_{−n}(x)Cν_{n}(x)^{θ} + (γγ̂)_{−n−1}(x)Cν_{n+1}(x)^{θ}

Since ν^{θ} < γγ̂, the sequence is Cauchy.

Let h_{∗}^{s} = lim_{n→∞} h_{n}^{s}_{∗}.

Step 3: the sequence T h_{n}^{s} : T W_{loc}^{c}(p) → T W^{c}(q) is Cauchy.

Since we have already found h_{∗}, it suffices to show that the sequence of functions v ↦ ∥T h_{n}v∥, for v ∈ P (T_{x}W_{loc}^{c}(p)), is Cauchy. This is uses a standard C^{1+θ} distortion estimate.

Lemma 1.3 There exists a K ≥ 1, such that, for all v ∈ P (T_{x}W_{loc}^{c}(p)) and n ≥ 1, we have:

K^{−1} ≤ \frac{Π_{k=0}^{n−1}∥T fv_{k}∥}{Π_{k=0}^{n−1}∥T fv′_{k}∥} ≤ K.

Proof. For w ∈ P (T M), let φ(w) = log ∥T fw∥. Note that φ is a δ-Hölder continuous function. Fix v ∈ P (T_{x}W_{loc}^{c}(p)) and n ≥ 1. Then

| log \frac{Π_{k=0}^{n−1}∥T fv_{k}∥}{Π_{k=0}^{n−1}∥T fv′_{k}∥}| = | ∑_{k=0}^{n−1} φ(v_{k}) − φ(v′_{k})|

≤ ∑_{k=0}^{n−1} Hd(v_{k}, v′_{k})^{δ}

≤ H ∑_{k=0}^{n−1} C^{δ}ν_{k}(x)^{θδ}
≤ HC^{δ}(1 − \overline{ν}^{θδ})^{−1},

where \overline{ν} = sup_{z∈M} ν(z) < 1. Setting K = exp(HC^{δ}(1 − \overline{ν}^{θδ})^{−1}) completes the proof. ⋄ Using this lemma, we fix v ∈ P (T_{x}W_{loc}^{c}(p)) and n ≥ 1 and estimate:

|∥T h_{n}^{s}v∥ − ∥T h_{n+1}^{s}v∥| = \frac{Π_{k=0}^{n−1}∥T fv_{k}∥}{Π_{k=0}^{n−1}∥T fv′_{k}∥}|∥T πv_{n}∥ − ∥T (fπf^{−1})v_{n}∥|
≤ C∥T πv_{n} − T (fπf^{−1})v_{n}∥
≤ C∥T πv_{n} − v_{n}∥ + ∥T (ff^{−1})v_{n} − T (fπf^{−1})v_{n}∥
≤ C′ν_{n}(x)^{θ}

Hence the sequence ∥T h_{n}^{s}∥ : P (T W_{loc}^{c}(p)) → R is Cauchy. Let T h = lim_{n→∞} T h_{n}^{s}.

Step 4: the function T h^{s} : T W_{loc}^{c}(p) → T W^{c}(q) is Hölder continuous.

We first show that h_{∗}^{s} : P (T W_{loc}^{c}(p)) → P (T W^{c}(q)) is Hölder continuous, uniformly in p ∈ M and q ∈ W^{c}(p, 1). Choose ε > 0 so that ν^{θ} < (γγ̂)^{1+ε}, and let ρ = (γγ̂)^{(2+ε)/θ}. Suppose that p′ ∈ W_{loc}^{c}(p). Then there exists an n ≥ 1 such that min{d(p, p′), 1} ∈ [ρ_{n}(p), ρ_{n−1}(p)).

Our previous calculations show that there exists a K > 0 such that

d_{C^{0}}(h_{∗}^{s}, h_{n}^{s}_{∗}) ≤ Kν_{n}^{θ}(γγ̂)_{−n}.

Let β = \frac{θε}{2+ε}, and note that

ν^{θ}(γγ̂)_{−1} < ρ^{β},

for all n. Thus it suffices to show that

d(h_{n}_{∗}(p, ξ), h_{n}_{∗}(p′, ξ)) ≤ O(ρ_{n}(p)^{β}),

for all n ≥ 1, p ∈ M and p′ ∈ W^{c}(p, ρ_{n}(p)).

Lemma 1.1 implies that

d(f_{∗}^{n}(p, ξ), f_{∗}^{n}(p′, ξ)) ≤ (γγ̂)_{−n}(p_{n})ρ_{n}(p)^{θ} < (γγ̂)_{n}(p)^{1+ε}.

Now property 2. of π^{s} implies that

d(π^{s}f_{∗}^{n}(p, ξ), π^{s}f_{∗}^{n}(p′, ξ)) ≤ d(π^{s}f_{∗}^{n}(p, ξ), f_{∗}^{n}(p, ξ)) + d(f_{∗}^{n}(p, ξ), f_{∗}^{n}(p′, ξ))
+d(f_{∗}^{n}(p′, ξ), π^{s}f_{∗}^{n}(p′, ξ))
≤ Cd(p_{n}, π^{s}(p_{n}))^{θ} + d(f_{∗}^{n}(p, ξ), f_{∗}^{n}(p′, ξ)) + Cd(p′_{n}, π^{s}(p′_{n}))^{θ}
≤ Cν_{n}(p)^{θ} + (γγ̂)_{n}(p)^{1+ε} + Cν_{n}(p′)^{θ}
≤ O((γγ̂)_{n}(p)^{1+ε})

Again applying Lemma 1.1, we obtain that

d(h_{n}_{∗}(p, ξ), h_{n}_{∗}(p′, ξ)) ≤ (γγ̂)_{−n}(p′_{n})O((γγ̂)_{n}(p)^{1+ε})
≤ O((γγ̂)_{n}(p)^{ε})
= O(ρ_{n}(p)^{β}).

We now show that log ∥T h^{s}∥ : P (T W_{loc}^{c}(p)) → R_{+} is Hölder continuous. Let v = (p, ξ) and w = (p′, ξ), and suppose as above that min{d(p, p′), 1} ∈ [ρ_{n}(p), ρ_{n−1}(p)). We showed in Step 3 that

log ∥T h^{s}(v)∥ = ∑_{i=0}^{∞} φ(v_{i}) − φ(h_{∗}^{s}(v_{i})),

where φ = log ∥T f∥. From this it follows that

|log ∥T h^{s}(p, ξ)∥ − log ∥T h^{s}(p′, ξ)∥| = |∑_{i=0}^{∞} (φ(v_{i}) − φ(h_{∗}^{s}v_{i})) − ∑_{i=0}^{∞} (φ(w_{i}) − φ(h_{∗}^{s}w_{i}))|
≤ ∑_{i=0}^{n−1} |φ(v_{i}) − φ(w_{i})| + ∑_{i=0}^{n−1} |φ(h_{∗}^{s}v_{i}) − φ(h_{∗}^{s}w_{i})|

+ ∑_{i=n}^{∞} |φ(v_{i}) − φ(h_{∗}^{s}v_{i})| + ∑_{i=n}^{∞} |φ(w_{i}) − φ(h_{∗}^{s}w_{i})|

≤ O(∑_{i=0}^{n−1} d(v_{i}, w_{i})^{δ} + ∑_{i=0}^{n−1} d(h_{∗}^{s}v_{i}, h_{∗}^{s}w_{i})^{δ}

+ ∑_{i=n}^{∞} d(v_{i}, h_{∗}^{s}v_{i})^{δ} + ∑_{i=n}^{∞} d(w_{i}, h_{∗}^{s}w_{i})^{δ})

≤ O(∑_{i=0}^{n−1}((γγ̂)_{−i}ρ_{n}^{θ})^{δ} + ∑_{i=0}^{n−1}((γγ̂)_{−i}ρ_{n}^{θ})^{βδ}

+ ∑_{i=n}^{∞} ν_{i}^{θδ})

≤ O (((γγ̂)_{−n}ρ_{n}^{θ}))^{βδ} + ν_{n}^{θδ})

Recall that we have chosen n so that

d(v_{n}, w_{n}) ≤ (γγ̂)_{−n}(p_{n})ρ_{n}(p)^{θ} < (γγ̂)_{n}(p)^{1+ε} = ρ_{n}(p)^{θ(1+ε)/(2+ε)}.

We also have that

ν^{θ} < (γγ̂)^{1+ε}

From these facts we conclude that

|log ∥T h^{s}(p, ξ) − log ∥T h^{s}(p′, ξ)∥| ≤ O(ρ_{n}(p)^{β′}),

where

β′ = δβ \frac{θ(1 + ε)}{2 + ε} = \frac{δθε(1 + ε)}{(2 + ε)^{2}}.

References

[BPS] Barreira, L., Pesin, Ya., and J. Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math. (2) 149 (1999), no. 3, 755–783.

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

[BW] Burns, K. and A. Wilkinson, On the ergodicity of partially hyperbolic systems, preprint.

[PR] Pinto, A. A. and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics. Bull. London Math. Soc. 34 (2002), no. 3, 341–352.

[PW] Pollicott, M., and P. Walkden, Livšic theorems for connected Lie groups, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2879–2895.

[PSW] Pugh, C., Shub, M., and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), no. 3, 517–546