Nonexistence of SBR Measures for some Diffeomorphisms that are “Almost Anosov”

Huyi Hu and Lai-Sang Young*

Department of Mathematics, University of Arizona, Tucson, AZ 85721

Abstract. The purpose of the paper is to present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Each example has a fixed point at which the larger eigenvalue is equal to one and the smaller eigenvalue is less than one.

§0 Introduction

Let f : M → M be a C^{2} Anosov diffeomorphism of a compact connected Riemannian manifold, and let m denote the Riemannian measure on M. A result of Sinai (see e.g. [S]) says that f admits a unique invariant Borel probability measure μ with the property that μ has absolutely continuous conditional measures on unstable manifolds. This is the invariant measure that is observed physically, for if ϕ : M → IR is a continuous function, then for m -a.e. x ∈ M,

\frac{1}{n} ∑_{i=0}^{n−1} ϕ(f^{i}x) → ∫ ϕdμ

as n → ∞. The dynamical system (f, μ) is “chaotic” in the following sense: it has positive Lyapunov exponents; its metric entropy is equal to the sum of its positive Lyapunov exponents; (f, μ) is measure-theoretically isomorphic to a Bernoulli shift; and it has exponential decay of correlations for Hölder continuous test functions. These results have been extended to Axiom A attractors by Bowen, Ruelle, etc. (See e.g. [B].)

In this article we will refer to an invariant measure having absolutely continuous conditional measures on unstable manifolds as a Sinai-Bowen-Ruelle measure or an SBR measure. The work of Oseledec, Pesin and others allows us to extend this notion to a nonuniform setting. (See [P] and [LS].) While some of the properties of SBR measures carry over (see e.g. [LY], Part I), the question of existence of SBR measures in this broader context remains poorly understood. We formulate this question more precisely: given a diffeomorphism which appears to be hyperbolic in a large part of phase space, can one decide whether or not it admits an SBR measure? So far there are very few results outside of Axiom A, and these results involve delicate estimates. See e.g. [BC], [BY] for results on the Hénon attractors.

The purpose of the paper is to present some very simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Precise statements of our results are given in §1. For now imagine slowly deforming a hyperbolic toral automorphism near the origin O until its derivative has one eigenvalue equal to 1 and the other eigenvalue less than 1. Our theorem says that for the resulting diffeomorphism,

* This research is partially supported by the National Science Foundation.

\frac{1}{n} ∑_{i=0}^{n−1} δ_{f^{i}x} → δ_{O} for almost every x with respect to Lebesgue measure. (δ_{x} denotes Dirac

measure at x.) This example can be made to be topologically conjugate to the original toral automorphism, and so it is topologically “chaotic”. From the statistical point of view, however, it is totally deterministic in the sense that for almost every initial condition, the trajectory spends nearly one hundred percent of its time arbitrarily near the origin O.

Our result can be thought of as a two dimensional version of the following result. Let f : [0, 1] → [0, 1] be a piecewise C^{2}, piecewise expanding map of the unit interval with f′ = 1 at a fixed point. Then f cannot admit a finite absolutely continuous invariant measure. (See [PI].) The two dimensional situation is, however, not entirely identical to that in one dimension, for clearly there exist area preserving diffeomorphisms with positive Lyapunov exponents and nonhyperbolic fixed points. A more detailed analysis of whether or not systems that are “almost Anosov” can admit SBR measures will be given in a forthcoming paper by the first named author.

§1 Assumptions and Statements of Results

Let M be a C^{∞} two dimensional compact manifold without boundary, let m denote the Riemannian measure on M, and let f ∈ Diff^{2}(M). We assume throughout this paper that f satisfies the following two conditions.

Assumption I.

1. f has a fixed point p, i.e. fp = p.

2. There exist a constant κ^{s} < 1, a continuous function κ^{u} with

κ^{u}(x) \{ \begin{matrix}= 1, & at x = p , \\ > 1, & elsewhere, \end{matrix}

and a decomposition of the tangent space T_{x}M at every x ∈ M into

T_{x}M = E_{x}^{u} ⊕ E_{x}^{s}

such that

∥Df_{x}v∥ ≤ κ^{s}∥v∥, ∀v ∈ E_{x}^{s},
∥Df_{x}v∥ ≥ κ^{u}(x)∥v∥, ∀v ∈ E_{x}^{u},

and

∥Df_{p}v∥ = ∥v∥, ∀v ∈ E_{p}^{u}.

Assumption II. f is topologically transitive on M.

Definition 1.1. An f−invariant Borel probability measure μ on M is called an SBR measure for f : M → M if

i) (f, μ) has positive Lyapunov exponents almost everywhere;

ii) μ has absolutely continuous conditional measures on unstable manifolds.

We give the precise meaning of the second condition above.

Let ξ be a measurable partition of a measure space X, and let ν be a probability measure on X. Then there is a family of probability measures {ν_{x}^{ξ} : x ∈ X} with ν_{x}^{ξ}(ξ(x)) = 1 such that for every measurable set B ⊂ X, x → ν_{x}^{ξ}(B) is measurable and

νB = ∫_{X} ν_{x}^{ξ}(B)dν(x).  (1.1)

The family {ν_{x}^{ξ}} is called a canonical system of conditional measures for ν and ξ. (For a reference, see e.g. [R].)

Suppose now that f : (M, μ) → (M, μ) has positive Lyapunov exponents almost everywhere. Then for a.e. x, the unstable manifold W^{u}(x) exists and is an immersed submanifold of M (see [P]). A measurable partition ξ of M is said to be subordinate to unstable manifolds if for μ -a.e. x, ξ(x) ⊂ W^{u}(x) and contains an open neighborhood of x in W^{u}(x). Let m_{x}^{u} denote the Riemannian measure induced on W^{u}(x). We say that μ has absolutely continuous conditional measures on unstable manifolds if for every measurable partition ξ that is subordinate to unstable manifolds, μ_{x}^{ξ} is absolutely continuous with respect to m_{x}^{u} (written μ_{x}^{ξ} << m_{x}^{u}) for μ -a.e. x ∈ M. (For more details, see e.g. [LS]).

It is easy to verify that if μ_{x}^{ξ} << m_{x}^{u} for one measurable partition ξ subordinate to unstable manifolds, then the same property holds for all other measurable partitions subordinate to unstable manifolds.

We now state our results. Let f and M be as in the beginning of this section.

Theorem A. f does not admit SBR measures.

Theorem B. For m -a.e. x ∈ M,

lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} δ_{f^{i}x} = δ_{p}

where δ_{z} is the Dirac measure at z, and the above convergence is in the weak ^{∗} topology.

As a by-product of our proofs for Theorem A and Theorem B, we obtain the following. Theorem C. f has an infinite invariant measure μ̄ with the following properties:

(i) if U is any open neighborhood of p in M, then μ̄(M\U) < ∞; (ii) μ̄ has absolutely continuous conditional measures on weak unstable manifolds.

Remark 1.2. Weak unstable manifolds are defined in Proposition 2.2 (2). Note that the definition of absolutely continuous conditional measures on unstable manifolds makes sense even though μ̄ is a σ−finite measure.

One could think of μ̄ as an infinite SBR measure. In this paper, however, the term “SBR measure” without any qualifications will always be reserved for probability measures.

§2. Preliminaries

Lemma 2.1. The maps x → {E_{x}^{u}} and x → {E_{x}^{s}} are continuous.

This is an easy consequence of the “gap” between κ^{s} and 1 = inf{κ^{u}(x) : x ∈ M}. The proof is left to the reader.

We will use the following notation: for β > 0, E_{x}^{u}(β) = {v ∈ E_{x}^{u} : |v| ≤ β}, E_{x}^{s}(β) = {v ∈ E_{x}^{s} : |v| ≤ β}, and E_{x}(β) = E_{x}^{u}(β) × E_{x}^{s}(β).

Proposition 2.2. There exist two continuous foliations F^{u} and F^{s} on M tangent to E^{u} and E^{s} respectively for which the following hold.

(1) The leaf of F^{s} through x, denoted by F^{s}(x), is the stable manifold at x, i.e.

F^{s}(x) = W^{s}(x) = {y ∈ M : ∃C = C_{y}, s.t. d(f^{n}x, f^{n}y) ≤ C(κ^{s})^{n} ∀n ≥ 0}.

(2) The leaf of F^{u} through x, denoted by F^{u}(x), is the unstable or “weak unstable” manifold at x, i.e.

F^{u}(x) = {y ∈ M : lim_{n→∞} d(f^{−n}x, f^{−n}y) = 0}.

(3) There exist constants β > 0 and D > 0 such that for all x ∈ M, if F_{β}^{u}(x) is the component of F^{u}(x) ∩ exp_{x} E_{x}(β) containing x, then exp_{x}^{−1} F_{β}^{u}(x) is the graph of a function ϕ_{x}^{u} : E_{x}^{u}(β) → E_{x}^{s}(β) with ϕ_{x}^{u}(0) = 0 and ∥ϕ_{x}^{u}∥_{C^{2}} ≤ D. The analogous statement holds for F_{β}^{s}(x).

Proof: These results follow from Theorem 5.5 and Theorem 5A.1 in [HPS]. We indicate how F^{u} is obtained.

Let x ∈ M be fixed. If ϕ : E_{x}^{u}(β) → E_{x}^{s}(β) is a function with ϕ(0) = 0 and Lip(ϕ) ≤ 1, we let G_{x}ϕ : E_{fx}^{u}(β) → E_{fx}^{s}(β) be the function defined by

graph(G_{x}ϕ) = (exp_{fx}^{−1} ∘f ∘ exp_{x}(graph ϕ)) ∩ E_{fx}(β).

Note that G_{x}ϕ is always well defined, even though f is not uniformly hyperbolic. The function ϕ_{x}^{u} in assertion (3) is then obtained as the pointwise limit as n → ∞ of

G_{f^{−1}x} ∘ ⋯ ∘ G_{f^{−n+1}x} ∘ G_{f^{−n}x}(0),

where 0 represents the 0 function from E_{f^{−n}x}^{u}(β) to E_{f^{−n}x}^{s}(β), and F^{u}(x) is defined to be

⋃_{n≥0} f^{n}(F_{β}^{u}(f^{−n}x)).

Remark 2.3. For convenience we will write W^{u}(x) = F^{u}(x), W_{β}^{u}(x) = F_{β}^{u}(x) etc. and refer to W^{u}(x) and W_{β}^{u}(x) as the “unstable manifold” and “local unstable manifold” respectively at x, even though points on these manifolds may not be contracted exponentially in backwards time.

The Lipschitzness of the W^{s}−foliation will be very important for us later on. We give the form of the definition that will be used.

Definition 2.4. Let Σ_{1} and Σ_{2} be two W^{u}−leaves, and let θ : Σ_{1} → Σ_{2} be a continuous map defined by sliding along the W^{s}−leaves, i.e. for x ∈ Σ_{1}, θ(x) ∈ Σ_{2}∩W^{s}(x). We say W^{s} is Lipschitz if θ is Lipschitz for every (Σ_{1}, Σ_{2}; θ).

For y ∈ W^{s}(x), let d^{s}(x, y) denote the distance between x and y measured along W^{s}(x), and for z ∈ W^{u}(x), let d^{u}(x, z) be defined similarly.

Proposition 2.5. The W^{s}−foliation is Lipschitz. In fact, given D_{1} > 0, there exists L_{1} > 0 such that for every (Σ_{1}, Σ_{2}; θ) with d^{s}(x, θ(x)) < D_{1} ∀x ∈ Σ_{1}, the Lipschitz constant of θ is less than or equal to L_{1}.

Proof: This result follows from the stronger statement that the map x → E_{x}^{s} is C^{1}, which can be obtained using the same ideas in the proof of Theorem 6.3 in [HP]. We sketch a more direct proof here for the convenience of the reader.

Let x_{1} ∈ Σ_{1}, and let γ be an arbitrarily short segment in Σ_{1} containing x_{1}. We will argue that l(γ) ≈ l(θγ), where l denotes length and “≈” means “up to a constant”.

By taking a suitably large iterate of f, we may assume that f^{n}γ and f^{n}(θγ) are very near each other, and l(f^{n}γ) ≈ d^{s}(f^{n}x, f^{n}(θx)) for x ∈ γ. Notice that ∀x ∈ γ, d^{s}(f^{n}(x), f^{n}(θx)) ≤ D_{1}(κ^{s})^{n} ∀n > 0. Also, |Df_{x}^{n}|_{E_{x}^{u}} | ≥ 1 ∀n ≥ 0. Observe the following:

(1) l(f^{n}γ) ≈ l(θ(f^{n}γ));

(2) |Df_{x}^{n}|_{E_{x}^{u}} | ≈ |Df_{θx}^{n}|_{E_{θx}^{u}} | ∀x ∈ γ;
(3) ∀y_{1}, y_{2} ∈ γ,

\frac{|Df_{y_{1}}^{n}|_{E_{y_{1}}^{u}} |}{|Df_{y_{2}}^{n}|_{E_{y_{2}}^{u}} |} ≈ ∏_{i=0}^{n−1}(1 ± const · d(f^{i}y_{1}, f^{i}y_{2})) ≤ (1 ± const · l(f^{n}γ))^{n}
≈(1 ± const · D_{1}(κ^{s})^{n})^{n} ≈ const.

(The proof of (3) uses the boundness of the C^{2} norms of ϕ_{x}^{u}. See Proposition 2.2.3).) Combining (1)-(3), we get l(γ) ≈ l(θγ).

Lemma 2.1 and Proposition 2.2 imply that f has a local product structure, i.e. there exist constants 0 < ϵ < β such that ∀y, z ∈ M with d(y, z) < ϵ, [y, z] := W_{β}^{u}(y) ∩ W_{β}^{s}(z) and [z, y] := W_{β}^{u}(z) ∩ W_{β}^{s}(y) each contains exactly one point.

A rectangle R is a set in M such that y, z ∈ R implies [y, z], [z, y] ∈ R. If γ^{u}, γ^{s} are segments of W^{u}− and W^{s}−leaves respectively, then [γ^{u}, γ^{s}] denotes the rectangle {[y, z] : y ∈ γ^{u}, z ∈ γ^{s}} provided that everything makes sense. If R is a rectangle and x ∈ R, we let W^{u}(x, R) = W_{β}^{u}(x) ∩ R and W^{s}(x, R) = W_{β}^{s}(x) ∩ R. If Q and R are two rectangles, we say that f^{n}Q u−crosses R if ∀x ∈ Q with f^{n}x ∈ R, f^{n}W^{u}(x, Q) ∩ R = W^{u}(f^{n}x, R).

We record a simple fact that will be used in §4.

Proposition 2.6. W^{u}(p) and W^{s}(p) are both dense in M.

Proof: We only prove the proposition for W^{s}(p).

Let P be a rectangle containing p and let R be any other rectangle, both with nonempty interiors. Let R̂ be a strictly smaller rectangle lying in the interior of R. By the topological transitivity of f, ∃n > 0 such that f^{n}R̂ ∩ P ≠ ∅. For n sufficiently large, f^{n}R is considerably longer than f^{n}R̂ in the u−direction. We may therefore assume that f^{n}R u−crosses P. This implies that f^{−n}W^{s}(p, P) ∩ R ≠ ∅.

§3. Distortion Estimates

The goal of this section is to prove the following.

Proposition 3.1. Given any small rectangle P containing p in its interior, there exist constants δ > 0 and J > 1 such that if γ is a W^{u}−segment with l(γ) ≤ δ and γ ∩ P = ∅, then ∀y, z ∈ γ and n > 0,

J^{−1} ≤ \frac{|Df_{z}^{−n}|_{E_{z}^{u}} |}{|Df_{y}^{−n}|_{E_{y}^{u}} |} ≤ J.

We write y_{i} = f^{−i}y, z_{i} = f^{−i}z and γ_{i} = f^{−i}γ. If Γ is a W^{u}−segment in P, let Γ̃ = {[p, x] : x ∈ Γ}. In what follows, the letter C will be used to denote a generic constant, which is allowed to depend only on f.

Lemma 3.2. Let P^{+} be one of the components of fP \P, and let γ = W^{u}(x, P^{+}) for some x ∈ P^{+}. Assume that γ_{i} ⊂ P for i = 1, ⋯ n − 1. Then for any y, z ∈ γ,

log \frac{|Df_{z}^{−n}|_{E_{z}^{u}} |}{|Df_{y}^{−n}|_{E_{y}^{u}} |} ≤ C \frac{d^{u}(y, z)}{l(γ)}.

Proof: First, we have for j ≤ n,

log \frac{|Df_{z}^{−j}|_{E_{z}^{u}} |}{|Df_{y}^{−j}|_{E_{y}^{u}} |} = log ∏_{i=0}^{j−1}(1 + \frac{|Df_{z_{i}}^{−1}|_{E_{z_{i}}^{u}} − Df_{y_{i}}^{−1}|_{E_{y_{i}}^{u}} |}{|Df_{y_{i}}^{−1}|_{E_{y_{i}}^{u}} |}) ≤ C ∑_{i=0}^{j−1}|Df_{z_{i}}^{−1}|_{E_{z_{i}}^{u}} − Df_{y_{i}}^{−1}|_{E_{y_{i}}^{u}} |.

Using Proposition 2.2 (3) and the Lipschitzness of W^{s} (Proposition 2.5), we see that

|Df_{z_{i}}^{−1}|_{E_{z_{i}}^{u}} − Df_{y_{i}}^{−1}|_{E_{y_{i}}^{u}} | ≤ Cd^{u}(z_{i}, y_{i}) ≤ Cl(γ_{i}) ≤ Cl(γ̃_{i}).

Since the γ̃_{i} are pairwise disjoint, we have

∑_{i=1}^{j−1} l(γ̃_{i}) ≤ l(W^{u}(p, P)).

The arguments above tell us that ∀j ≤ n,

\frac{d^{u}(z_{j}, y_{j})}{l(γ_{j})} ≤ C \frac{d^{u}(z, y)}{l(γ)}.

We conclude that

log \frac{|Df_{z}^{−n}|_{E_{z}^{u}} |}{|Df_{y}^{−n}|_{E_{y}^{u}} |} ≤ C ∑_{j=0}^{n−1} d^{u}(z_{j}, y_{j}) ≤ C \frac{d^{u}(z, y)}{l(γ)}.

Proof of Proposition 3.1:

Let 0 = n_{0} = n_{0} + k_{0} ≤ n_{1} < n_{1} + k_{1} < ⋯ < n_{t} < n_{t} + k_{t} ≤ n_{t+1} = n be such that

γ_{j} ∩ P ≠ ∅ ∀n_{i} < j < n_{i} + k_{i} 1 ≤ i ≤ t,

and

γ_{j} ∩ P = ∅, otherwise.

Then

log \frac{|Df_{z}^{−n}|_{E_{z}^{u}} |}{|Df_{y}^{−n}|_{E_{y}^{u}} |} = ∑_{i=1}^{t} log \frac{|Df_{z_{n_{i}}}^{−k_{i}}|_{E_{zn_{i}}^{u}} |}{|Df_{y_{n_{i}}}^{−k_{i}}|_{E_{yn_{i}}^{u}} |} + ∑_{i=0}^{t} ∑_{j=n_{i}+k_{i}}^{n_{i+1}−1} log \frac{|Df_{z_{j}}^{−1}|_{E_{z_{j}}^{u}} |}{|Df_{y_{j}}^{−1}|_{E_{y_{j}}^{u}} |}.

Lemma 3.2 applied to the terms in the first series gives each a contribution of C′d^{u}(y_{n_{i}}, z_{n_{i}}), where C′ is a constant depending on P. This is summable since d^{u}(y_{n_{i}}, z_{n_{i}}) decreases exponentially in i. Each term in the second series is less than or equal to Cd^{u}(y_{j}, z_{j}), so we again have a geometric series.

§4 Proof of Theorem A

The following one-dimensional fact plays a key role.

Lemma 4.1. Let h : [−1, 1] → IR be a C^{2} map with h(0) = 0, h′(0) = 1 and h′(x) ≥ 1

∀x ∈ [−1, 1]. Let a_{0} ∈ [0, 1], and let a_{i} = h^{−i}a_{0} for i ≥ 1. Then ∑_{i=0}^{∞} a_{i} = ∞.

Proof: From the Taylor expansion of h, we know that for x > 0,

hx ≤ x + Lx^{2}  (∗)

for some L. Increasing L if necessary, we may assume that a_{1} ≥ \frac{1}{L}. We will show that a_{i} ≥ \frac{1}{Li} implies that a_{i+1} ≥ \frac{1}{L(i+1)}. Suppose this is not true. Then

a_{i+1} + La_{i+1}^{2} < \frac{1}{L(i + 1)}(1 + \frac{1}{i + 1}) = \frac{1}{Li}(1 − \frac{1}{i + 1})(1 + \frac{1}{i + 1}) < \frac{1}{Li} ≤ a_{i},

contradicting (∗) with x = a_{i+1}.

With a_{i} ≥ \frac{1}{Li} ∀i ≥ 1, the desired conclusion follows.

Before giving the proof of Theorem A, we recall some facts from general nonuniform hyperbolic theory. Let f be an arbitrary C^{1+α} diffeomorphism (not having anything to do with the situation in this paper), and suppose that f preserves an SBR measure μ. Let ξ be a partition subordinate to W^{u}. Then it is proved in [L] that for μ -a.e. x, the density ρ_{x} of μ_{x}^{ξ} with respect to m_{x} satisfies

\frac{ρ_{x}(z)}{ρ_{x}(y)} = \frac{∏_{i=0}^{∞}|Df_{z_{i}}^{−1}|_{E_{z_{i}}^{u}} |}{∏_{i=0}^{∞}|Df_{y_{i}}^{−1}|_{E_{y_{i}}^{u}} |}

for all y, z ∈ ξ(x). (In particular the quotient on the right makes sense.) The following is also known to be true. Let {Λ_{l}} be the “Pesin sets”, i.e. sets on which f^{n} has uniform estimates. Then for every l, ∃δ_{l} > 0 such that W_{δ_{l}}^{u}(x) exists for every x ∈ Λ_{l}. Also, ∃C_{l} > 0 such that ∀x ∈ Λ_{l},

C_{l}^{−1} ≤ \frac{ρ_{x}(z)}{ρ_{x}(y)} ≤ C_{l} ∀y, z ∈ W_{δ_{l}}^{u}(x).

(See e.g. [P].)

We now return to the situation considered in this paper, i.e. f is again assumed to satisfy Assumptions I and II.

Proof of Theorem A:

Suppose, to derive a contradiction, that f admits an SBR measure μ. Then there is a rectangle R ⊂ M such that μ(R ∩ Λ_{l}) > 0 for some l, and W^{u}(x, R) ⊂ W_{δ_{l}}^{u}(x) ∀x ∈ R ∩ Λ_{l}.

We fix a rectangle of the form P = [W_{δ}^{u}(p), W_{δ}^{s}(p)] and let Q = f^{−1}P \P. Let ξ be the partition of Q given by ξ(x) = W^{u}(x, Q). An argument similar to that in Proposition 2.6 shows that f^{n}R u−crosses Q for some n > 0. It follows from our discussion above that there is a set Q̂ ⊂ Q ∩ f^{n}R with μQ̂ > 0 such that

(i) x ∈ Q̂ ⟹ ξ(x) ⊂ Q̂; and

(ii) ∃C_{0} > 0 such that ∀x ∈ Q̂,

C_{0}^{−1} ≤ \frac{ρ_{x}(z)}{ρ_{x}(y)} ≤ C_{0} ∀y, z ∈ W^{u}(x, Q).

Let

Q^{(i)} = {y ∈ Q̂ : f^{j}y ∈ P for j = 1, ⋯, i},

and let γ̃^{(i)} be the projection of Q^{(i)} onto W^{u}(p, P) by sliding along W^{s}. Then the density estimate above together with the Lipschitzness of the W^{s}−foliation gives

μQ^{(i)} ≥ Cl(γ̃^{(i)}).

Using the facts that f^{i}Q^{(i)}, i = 1, 2, ⋯, are pairwise disjoint subsets of P, and μ is an invariant measure, we have

μP ≥ ∑_{i=1}^{∞} μ(f^{i}Q^{(i)}) = ∑_{i=1}^{∞} μQ^{(i)} ≥ C ∑_{i=1}^{∞} l(γ̃^{(i)}).

Lemma 4.1 applied to f|_{W_{β}^{u}}(p) tells us that this sum diverges, contradicting μ(M) = 1.

§5 Proofs of Theorems B and C

We first construct some neighborhoods of p that are convenient to work with. For a closed rectangle R, let ∂^{s}R = {x ∈ R : x ∉ int W^{u}(x, R)}.

Lemma 5.1. There exist rectangles P of the form P = [γ^{u}, γ^{s}] satisfying the follows:

(1) γ^{u} and γ^{s} are segments of W^{u}(p) and W^{s}(p) respectively such that p lies in the interiors of γ^{u} and γ^{s};

(2) there is a compact segment Ŵ^{s} ⊂ W^{s}(p) with fŴ^{s} ⊂ Ŵ^{s} such that ∂^{s}P ⊂ Ŵ^{s} and (Ŵ^{s}\W^{s}(p, P)) ⋂ int P = ∅;

(3) there is a compact segment Ŵ^{u} ⊂ W^{u}(p) with analogous properties. Moreover, the diameter of P can be chosen arbitrarily small.

Proof: Use the fact that W^{u}(p) and W^{s}(p) are dense in M. See Lemma 2.6.

We fix P as above, and consider the first return map g : M\P → M\P. That is, for x ∈ M\P, if τ(x) is the smallest positive integer with f^{τ(x)}x ∈ M\P, then gx = f^{τ(x)}x. Note that g is not defined on a set of Lebesgue measure 0 on M\P, but this will not concern us.

Lemma 5.2. There exists a g−invariant Borel probability measure μ with the property that μ has absolutely continuous conditional measures on the unstable manifolds of f.

Proof: Let L = W^{u}(x, P^{+}), where P^{+} is one of the components of fP \P and x ∈ P^{+}. Let m_{L} be the Lebesgue measure on L and let g_{∗}^{n}m_{L} be the push-forward of m_{L}, i.e., (g_{∗}^{n}m_{L})(E) = m_{L}(g^{−n}E). We may take μ to be any accumulation point in the weak ^{∗}

topology of \frac{1}{n} ∑_{i=0}^{n−1} g_{∗}^{i}m_{L}.

The g−invariance of μ is clear. To show that it has absolutely continuous conditional measures, it suffices to consider rectangles R in M\P with small diameters. We assume also that int R ∩ Ŵ^{s} = ∅, where Ŵ^{s} is as in Lemma 5.1. The significance of the second condition is as follows: g^{i}L is the disjoint union of W^{u}−segments, each one of which begins and ends at some point in g^{j}Ŵ^{s} for some j ≥ 0. Since fŴ^{s} ⊂ Ŵ^{s}, each component of g^{i}L that intersects R u−crosses R. Let ρ_{i} denote the density of g_{∗}^{i}m_{L} with respect to Lebesgue measure on g^{i}L. Then by Proposition 3.1, ∃J > 0 (independent of i) such that for all x, y in the same component of g^{i}L ∩ R,

J^{−1} ≤ \frac{ρ_{i}(x)}{ρ_{i}(y)} ≤ J.

This bound on densities is passed on to the limit measure μ.

Proof of Theorem C:

Let Q_{i} = {x ∈ M\P : τ(x) = i}, where τ is the return time to M\P. Define

μ̄ = ∑_{i=1}^{∞} ∑_{j=0}^{i−1} f_{∗}^{j}(μ|_{Q_{i}}).

Then μ̄ is clearly f−invariant.

Let U be a neighborhood of p. Then M\U is contained in M\( ⋂_{i=−n}^{n} f^{i}P ) for some

large n, and this latter set clearly has finite μ̄−measure. This proves that μ̄ is at most σ−finite. It cannot be finite because it has absolutely continuous conditional measures on unstable manifolds, and Theorem A says that f does not admit SBR measures.

Recall that m denotes the Lebesgue measure on M. We now study the asymptotic behavior of trajectories starting at x for m -a.e. x.

Lemma 5.3. Let g and μ be as in Lemma 5.2. Then (g, μ) is ergodic.

Proof: We will follow the two standard steps in the proof of ergodicity of SBR measures for hyperbolic systems without discontinuities. The first step is to use Hopf’s argument to show that given a rectangle R, m -a.e. x ∈ R is “future-generic” with respect to some ergodic measure μ_{R}, with μ_{R} possibly depending on R. (“Future-genericity” means

that \frac{1}{n} ∑_{i=0}^{n−1} ϕ ∘ g^{i}(x) → ∫ ϕdμ_{R} as n → +∞ for every continuous function ϕ : M → R.

“Past-genericity” is defined similarly.) The second step is to show that μ_{R} = μ for all R.

Let R ⊂ M\P be a rectangle. Note that when we use the word “rectangle” or the symbol “W^{u}(x, R)” in this paper, we are always referring to the stable and unstable manifolds of f — which are not necessarily stable and unstable manifolds of g! First we need to argue that for suitable R, W^{u}(x, R) is indeed a local unstable manifold of g, in the sense that ∀y ∈ W^{u}(x, R), d(g^{−n}x, g^{−n}y) → 0 as n → ∞. This is true if int R ∩ Ŵ^{s} = ∅, for this condition will guarantee that ∀n ≥ 0, f^{−n}W^{u}(x, R) is either entirely in P or it does not intersect P. Similarly, W^{s}(x, R) is a local stable manifold of g if R ∩ Ŵ^{u} = ∅.

We recall Hopf’s argument (see, e.g. [A]) for a rectangle R with the properties in the last paragraph. Since the conditional measures of μ are absolutely continuous on unstable manifolds (Lemma 5.2), there exists L = W^{u}(x, R) such that m_{x}^{u} -a.e. y ∈ L is generic (both future and past) with respect to some ergodic measure ν_{y}. All the ν_{y}’s are in fact identical because as n → +∞, d(g^{−n}y, g^{−n}z) → 0 ∀y, z ∈ L. We call this common measure μ_{R}. Now if y is future generic with respect to μ_{R}, then z is future generic with respect to μ_{R} ∀z ∈ W^{s}(y, R). It then follows from the Lipschitzness of the W^{s}−foliation (Proposition 2.5) that m -a.e. z ∈ R is future generic with respect to μ_{R}.

To carry out the second step, it suffices to observe that if R_{1} and R_{2} are rectangles with the properties above, then ∃n > 0 such that g^{n}R_{1} ∩ R_{2} ≠ ∅.

Proof of Theorem B:

We will show that given arbitrarily small numbers α > 0 and ϵ > 0, there exist neighborhoods P_{2} ⊂ P_{1} of p with diam P_{1} ≤ α such that for m -a.e. x ∈ M\P_{2},

\frac{#\{0 ≤ k ≤ n : f^{n}x ∈ M\P_{1}\}}{#\{0 ≤ k ≤ n : f^{n}x ∈ P_{1}\P_{2}\}} < ϵ

for all sufficient large n.

To see this, let P_{1} be a rectangle of the type in Lemma 5.1. Let g_{1} : M\P_{1} → M\P_{1} and μ_{1} be as in Lemma 5.2, and let μ̄ be the infinite measure in the proof of Theorem C. Let P_{2} be chosen small enough that μ̄(M\P_{1}) ≤ ϵμ̄(P_{1}\P_{2}). Then μ_{2} := μ̄|_{M\P_{2}} is invariant under the first return map g_{2} : M\P_{2} → M\P_{2}, and (g_{2}, μ_{2}) is ergodic. The Birkhoff Ergodic Theorem applied to (g_{2}, μ_{2}) completes the proof.

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