STATISTICAL PROPERTIES OF SOME ALMOST HYPERBOLIC SYSTEMS

HUYI HU

Abstract. Almost hyperbolic systems are smooth dynamical systems that are hyperbolic everywhere except for a finite set of points. Evidences from low-dimensional cases show that the topological properties of such systems are similar to that of uniformly hyperbolic systems, and the ergodic properties may be quite different. It admit SRB measures or infinite SRB measures. In the latter case, the systems are statistically deterministic in the sense that almost every orbit spends one hundred percent of its time arbitrarily close to the indifferent fixed points, though they are still topologically mixing. Even in the former case, correlation decay may change from exponential to power law.

0. Introduction

A smooth dynamical system is almost hyperbolic if it is hyperbolic everywhere except at a finite set of points. Ergodic properties of an almost hyperbolic system may be quite different from that of a uniformly hyperbolic system, even though we can make them topologically conjugate.

Let us imagine that we slowly deform a hyperbolic toral automorphism near the origin until the differential becomes identity. For some deformation the system does not admit any Sinai-Ruelle-Bowen measure. Instead, it admits a σ-finite measure that has absolutely continuous conditional measures on unstable manifolds. We call it an infinite SRB measure. For some other deformation, the resulting system still admits an SRB measure. However, the rate of decay of correlations of the system changes from exponential to polynomial. Similar phenomena also exist in some expanding maps with indifferent fixed points.

These properties imply different statistical behavior for the system. If the system has an infinite SRB measure, then for Lebesgue almost every initial condition, the orbit spends nearly one hundred percent of its time arbitrarily close to the indifferent fixed point, though it is also true that almost every orbit is dense on the torus. Hence, the system is deterministic from statistical point of view, and chaotic from topological point of view. The origin is a saddle, but statistically it looks like a sink. For the case that the system admits an SRB measure, the orbits visit every open set with positive frequency. However, they spend longer time near the indifferent fixed point, since the density of the measure is unbounded there. Therefore, the rate of mixing of the system, if measured by the rate of decay of correlations, is slower. So we can think that the system is “less chaotic” than the undeformed system.

1991 Mathematics Subject Classification. Primary 37A25; Secondary 37D25, 37C40.

Key words and phrases. Almost hyperbolic system, almost Anosov system, SRB measure, infinite SRB measure, decay of correlations.

The work is partially supported by the National Science Foundation grant #DMS-9970646; .

The behavior of such systems mainly depends on the distribution of the physically observable measure μ in the neighborhoods of the indifferent fixed point. More precisely, it depends on how fast the measure escape from the neighborhoods. Let

P_{n} be the sets of points that escape from a neighborhood at time n. If ∑_{n=1}^{∞} μP_{n}

diverges, then the system has an infinite SRB measure, and it is statistically de-

terministic. If ∑_{n=1}^{∞} μP_{n} converges, but μP_{n} decrease in a polynomial rate, then the

system has an SRB measure and has polynomial decay of correlations. This is the “less chaotic” case. If μP_{n} decrease exponentially fast, then the system has an SRB measure and has exponential decay of correlations. This happens in uniformly hyperbolic systems and some other nonuniformly hyperbolic systems people have studied.

Sometimes the information can be observed directly from the dynamics. Let us consider a piecewise smooth almost expanding map with an indifferent fixed point p at which | det Df_{p}| = 1. The potential functions ϕ we are interested in (for example, ϕ = − log | det Df|,) is negative and generically is of order n^{−1} on P_{n}. If on P_{n}, ϕ ∼ −(β + 1)/n, then the absolutely continuous invariant measure μ is infinite for 0 < β ≤ 1, and is finite with O(n^{−(β−1)}) as the rate of decay of correlations for β > 1. The systems we will discuss in Theorem 3.11, Fact 3.14 and Theorem 3.17 belong to the case. However, β may vary with x on P_{n} for systems in higher dimensional spaces. Therefore more detailed analysis is required.

In this paper we mainly consider almost Anosov systems on the surfaces and almost expanding maps on the intervals. We will describe the local behavior of the systems near the indifferent fixed points, and introduce the main idea for existence of SRB measures and rates of decay of correlations. Similar results should hold in general almost hyperbolic systems, however, we don’t know very much beyond these maps.

1. Definitions

We start with hyperbolic case. We define the rank of a cone in ℝ^{n} as the maximal dimension of a linear subspace contained in the cone. A pair of cones is called complete if the sum of their ranks is n.

Definition 1.1. Let f be a C^{r}, r > 1, diffeomorphism from a manifold M to itself. A closed invariant subset Λ ⊂ M is called an almost hyperbolic set, if there exist two continuous families of complete closed cones C_{x}^{u} and C_{x}^{s} on the tangent bundle T_{Λ}M such that except at a finite set S ⊂ Λ,

i) (invariance) Df_{x}C_{x}^{u} ⊂ int C_{fx}^{u}, and Df_{x}C_{x}^{s} ⊃ int C_{fx}^{s}; ii) (hyperbolicity) |Df_{x}(v)| > |v|, |Df_{x}^{n}(v)| → ∞ ∀v ∈ C_{x}^{u},

and |Df_{x}(v)| < |v|, |Df_{x}^{n}(v)| → 0 ∀v ∈ C_{x}^{s}.

If Λ = M, then we call the system an almost Anosov diffeomorphism.

It is possible that |Df_{x}(v)| = |v| for |v| ∈ C_{x}^{u} or C_{x}^{s} if x ∈ S. So the systems are not uniformly hyperbolic.

The reason that we use cones rather than decomposition T M = E^{u} ⊕ E^{s} is that the unstable and stable bundles may not be continuous. In uniformly hyperbolic systems, continuity of x → C_{x}^{u} and C_{x}^{s} follows from invariance and hyperbolicity. If we allow Df_{p} = id for some p ∈ S, then the cone families are not necessary continuous. In fact, the curvature of unstable and stable leaves depends on hyperbolicity. Near these points, the leaves can have a u-turn. This happens in an example constructed by A. Katok [K] of a C^{∞} Bernoulli diffeomorphism on the 2-dimensional sphere that is hyperbolic everywhere except at four points. In [GK] M. Gerber and A. Katok constructed a C^{∞} smooth model for pseudo-Anosov homeomorphisms in which there are more than one unstable and stable directions near the exceptional points. (See also [G] for analytic models.) These are impossible if the cone families continuous at every point in the manifolds.

One may also consider the case that S is a submanifold of M. (See [N] for a generalization of Theorem 3.8 to the case.)

For noninvertible systems, we consider piecewise almost expanding maps.

Definition 1.2. Let f be a C^{r}, r > 1, piecewise smooth map from a manifold M to itself, and Λ ⊂ M be a closed invariant subset. We say that f is almost expanding on Λ if f is uniformly expanding away from a finite set S. That is, for any ϵ > 0, there is κ > 1 such that if x ∈ Λ\B(S, ϵ), then

|Df_{x}(v)| > κ|v| ∀v.

Without loss generality, we assume that in both cases S is an f-invariant set. By considering f^{n}, we assume that S consists of fixed points. We are interested in the case that the fixed points are indifferent, because otherwise the systems become uniformly hyperbolic or uniformly expanding. Further, we assume that S consists of only one indifferent fixed point p except when otherwise stated.

Definition 1.3. A fixed point p is called indifferent if Df_{p} has an eigenvalue λ with |λ| = 1.

In this paper we only consider the systems that are topologically transitive, and allow Markov partitions R into finite number of elements {R_{j}}. We denote R_{n} =

⋁_{i=0}^{n−1} f^{−i}R, and then let R_{n}(x) ∈ R_{n} be the element that contains x. Restricted to

each element, f is C^{r} with r > 1. So the systems are smooth or piecewise smooth. We also assume that R_{0} is the element containing p. We denote

P_{n} = {x ∈ R_{0} : f^{k}x ∈ R_{0}, 0 ≤ k ≤ n − 1, f^{n}x ∉ R_{0}}, (1.1)

Q_{n} = ⋃_{i=0}^{n} P_{i} and O_{n} = ⋃_{i=n}^{∞} P_{i}. We will regard P_{0} = Λ\R_{0}

We will use the notation “∼” very often, where a(n) ∼ b(n) means that a(n) = b(n) + O(n^{−(1+t)}) for some t > 0.

2. Dynamics near Indifferent Fixed Points

Consider an almost expanding map f on the unit interval I. Assume that 0 is the fixed point and I_{0} is the element of the Markov partition that contains 0.

Take x ∈ I_{0}. Consider the backwards orbit {x_{n}}_{n=0}^{∞} of x, where x_{n} = f^{−n}x ∩ I_{0}. We know that if f′(0) > 1, then x_{n} converges to 0 exponentially fast. In our case, the rate is polynomial.

Fact 2.1. Suppose

fx = x(1 + x^{γ} + o(x^{γ})), x → 0. (2.1)

Then for any x ∈ I_{0}, the backwards orbit {x_{n}}_{0}^{∞} satisfies x_{n}^{γ} ∼ (γn)^{−1}.

Remark 2.2. By the fact we can see that on P_{n},

f′x ∼ 1 + \frac{1 + β}{n} , β = 1/γ. (2.2)

We cannot expect bounded distortion on any interval that contains 0, since (f^{n})′(0) = 1 and (f^{n})′(x_{n}) is unbounded by Fact 2.1 and (2.2). However, the following fact says that restricted to P_{k}, distortion is bounded. This is enough for us to get absolute continuity of invariant measures and to estimate the rates of decay of correlations.

Proposition 2.3. There exists constant J > 0 such that for any x, y ∈ P_{k}, for any n > 0, if x_{n} ∈ f^{−n}x and y_{n} ∈ f^{−n}y ∩ R_{n}(x_{n}), then

log \frac{|(f^{n})′(y_{n})|}{|(f^{n})′(x_{n})|} ≤ \frac{J}{x} d(x, y).

Now we consider almost hyperbolic systems on surfaces. Let p be an indifferent fixed point. If one of eigenvalues of Df_{p} is not equal to 1, then there is a gap between the spectrum of Df. So we can apply the stable manifold theorem ([HPS]) to obtain unstable and stable manifolds, which form C^{1} foliations. The latter means that the holonomy maps by sliding along the unstable or stable leaves are C^{1}. So near p, by choosing a suitable coordinate system, the map can be written as

f(x, y) = (x + ax^{2k+1} + h.o.t., λy + h.o.t. ), (2.3)

for some λ > 1, a < 0, or λ < 1, a > 0. Hence the orbits change polynomially in one direction and exponentially in the other direction. (Note that we need odd power for the second term in x-component to keep the map hyperbolic.)

For the case that 1 is the only eigenvalues of Df_{p}, there are two possibilities: the eigenspace of Df_{p} is one or two dimensional. In the first case the unstable and stable manifolds are tangent at p. These maps were studied by J. Lewowicz ([Lo]). Recently, it is proved by E. Catsigeras and H. Enrich [CE] that with some conditions on the coefficient of the Taylor expression, f have an SRB measure.

In the second case, which is simpler and has been understood better, Df_{p} = id. We assume the systems satisfy nondegenerate conditions below. It seems that similar results are also true if we use higher order jet instead of the third order one. However, there is no reference about it.

Definition 2.4. An almost Anosov diffeomorphism f : M → M is said to be nondegenerate, if there are constants κ^{+}, κ^{−} > 0 such that for all x near p,

|Df_{x}v| ≥ (1 + κ^{+}d(x, p)^{2})|v| ∀v ∈ C_{x}^{u},
|Df_{x}v| ≤ (1 − κ^{−}d(x, p)^{2})|v| ∀v ∈ C_{x}^{s} .

Note that Df_{p} = id by the assumption, D^{2}f_{p} = 0 by hyperbolicity, and D^{3}f_{p} ≠ 0 by the nondegenerate conditions. So near p the motion is dominated by D^{3}f_{p}, which makes the dynamics quite different (see [H1] for more details).

Since D^{3}f_{p} is a 3-form, the immediate consequence is that near p, f^{n^{2}} acts on x/n just like f acting on x up to a scale of n times. That is, f^{n^{2}}(tx/n) ∼ f(tx)/n, or

more precisely, lim_{t→0} \frac{f^{n^{2}}(tx/n)}{f(tx)/n} = 1. Df^{n^{2}} has the same property. We should recall that near a hyperbolic fixed point, the corresponding relation is f(tx/n) ∼ f(tx)/n, since f is close to a hyperbolic linear map. The facts are very helpful for us to understand the behavior of orbits when they are near p.

The system allows a decomposition of the tangent bundle into T M = E^{u} ⊕ E^{s}. For x ≠ p, existence and continuity of E_{x}^{u} and E_{x}^{s} can be obtained in the same way as for uniformly hyperbolic systems. Further, E_{x}^{u} and E_{x}^{s} satisfy the Hölder condition outside any neighborhood of p. (The Hölder coefficient increases as the size of the neighborhood shrinks.) We can also define E_{p}^{u} and E_{p}^{s}: they are the only lines such that D^{3}f_{p}(v, v, v) and v collinear for any v in the line. From the similarity between Df at x and Df^{n^{2}} at x/n, we can see that for any x, E_{tx}^{u} and E_{tx}^{s} has a limit as t → 0, and the limits are different if the directions of x are different. This is because near p, Df_{x} = id +(1/2)D^{3}f(x, x, ·), and D^{3}f(x, x, ·) change as the direction of x change, while near a hyperbolic fixed point, all Df_{x} are close to Df_{p}.

Note that between unstable and stable parts, there is no gap in the spectrum of Df. The stable manifold theorem doesn’t apply. However, (weak) unstable and stable manifolds can be obtained by graph transformation for x ≠ p. W_{loc}^{u}(p) and W_{loc}^{s}(p) can be defined as the set of points whose backward or forward orbits stay inside a small neighborhood of p. They are smooth curves and tangent to E_{p}^{u} and E_{p}^{s} respectively.

We have similar distortion estimates as for one dimensional case. The following result is proved in [H1] and refined in [H4].

Let P be a rectangle with p ∈ int P. Recall the definition of P_{n} in (1.1).

Proposition 2.5. There exists constant I > 0, depending on P, θ ∈ (0, 1) such that for any x, y ∈ P_{k}, k ≥ 0, with y ∈ W_{loc}^{u}(x),

log \frac{|Df_{y}^{−n}|_{E_{y}^{u}} |}{|Df_{x}^{−n}|_{E_{x}^{u}} |} ≤ Id^{u}(x, y)^{θ} ∀n > 0.

Away from p, the unstable and stable foliations are Lipschitz. That is, the holonomy maps π_{xy} : W_{loc}^{u(s)}(x) → W_{loc}^{u(s)}(y) are Lipschitz, if they stay away from W_{loc}^{u}(p) and W_{loc}^{s}(p). (Of cause, the Lipschitz constants increase as they get closer to p.) On the other hand, for any x close to p, the holonomy maps from W_{loc}^{u}(x) to W_{loc}^{u}(p) is only Hölder. Let us assume that in a neighborhood of p the map f can be written as the following in a local coordinate system,

f ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}x + x(ax^{2} + by^{2} + O((x^{2} + y^{2})^{3/2}) ) \\ y − y(cx^{2} + dy^{2} + O((x^{2} + y^{2})^{3/2}) ) \end{matrix} ) . (2.4)

Note that for any z ∈ W_{loc}^{u}(p), d(p, f^{−n}z) have the order n^{−1/2} by Fact 2.1.

Proposition 2.6. Let q ∈ W_{loc}^{s}(p) with q ≠ p. Then for any z ∈ W_{loc}^{u}(p), d(q, π_{pq}(f^{−n}z)) = O(n^{−b/2d}).

The non-Lipschitzness of the stable foliation is very important for the statistical behavior of the systems. It is d(q, π_{pq}(f^{−n}z)), not d(p, f^{−n}z), that plays the role, as we will see in Proposition 3.4 and Theorem 4.5. (If we construct a tower as in [Y1] or [Y2], then d(q, π_{pq}(f^{−n}z)) is the size of the n’th level.)

3. Invariant Measures

3.1. SRB Measures and Infinite SRB Measures. For an Anosov system f : M → 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 → ℝ is a continuous function, then for Lebesgue almost every point x ∈ M,

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

These results have been extended to Axiom A attractors by Ruelle, Bowen, etc. (See e.g. [B].) This measure is called an Sinai-Ruelle-Bowen measure or an SRB measure. Now there are some results about existence of SRB measures for nonuniformly hyperbolic systems such as Hénon maps and hyperbolic systems with singularities. (See [BY], [P], [JN], [BV], etc.)

Definition 3.1. An f-invariant probability measure μ is called an SRB measure for f : Λ → Λ if

i) (f, μ) has positive Lyapunov exponents almost everywhere; ii) μ has absolutely continuous conditional measures on unstable manifolds.

For the precise meaning of the second condition above, we refer [LS].

Some almost Anosov systems do not admit SRB measures. Instead, they admit σ-finite measures that have similar properties.

Definition 3.2. An f-invariant σ-finite measure μ is called an infinite SRB measure for f : Λ → Λ if μΛ = ∞, and for any R ⊂ Λ with μ(R) < ∞ and R = \overline{int R},

i) (f_{R}, μ_{R}) has positive Lyapunov exponents almost everywhere, and ii) μ_{R} has absolutely continuous conditional measures on unstable manifolds

of f. where f_{R} is the first return map of f on R, and μ_{R} is the conditional measure of μ on R.

In this paper, the term “SBR measure” without any qualifications will be reserved for probability measures.

We will see in Corollary 3.5 that orbit distributions are quite different between systems with SRB measures and those with infinite SRB measures.

For almost Anosov systems on compact surfaces, we have the following.

Theorem 3.3. Every nondegenerate almost Anosov diffeomorphism on a surface has either an SRB measure or an infinite SRB measure.

Proof. Take a suitable rectangle P that contains p as its interior point. Let g : M\P → M\P be the first return map.

Take a piece of unstable curve γ. Let m_{γ} be the Lebesgue measure on γ and let g_{∗}^{n}m_{γ} be the push-forward of m_{γ}, i.e., (g_{∗}^{n}m_{γ})(E) = m_{γ}(g^{−n}E). Then any accu-

mulation point μ̄ of n^{−1} ∑_{i=0}^{n−1} g_{∗}^{i}m_{γ} in the weak^{∗} topology is a g-invariant measure.

Let ρ_{i} denote the density of g_{∗}^{i}m_{γ} with respect to the Lebesgue measure on g^{i}γ. Then by distortion estimates in Proposition 2.5 for any small rectangle R ⊂ M\P, for all x, y in a same component of g^{i}γ ∩ R that crosses R,

e^{−I} ≤ \frac{ρ_{i}(x)}{ρ_{i}(y)} ≤ e^{I}. (3.1)

This bound on ρ_{i} is passed on to the limit measure μ̄. Then we get absolute continuity.

We can extend μ̄ to an f-invariant measure μ in an obvious way. If μ is finite, then we get an SRB measure (after normalization), otherwise we get an infinite SRB measure. □

To determine if μ is finite, we only need check if the return time on f^{−1}P \P is integrable, or equivalently, to see if μP is finite. The following proposition gives a direct way to estimate it, in which the notations are the same as in Proposition 2.6.

Proposition 3.4. Let q ∈ W_{loc}^{s}(p) with q ∈ f^{−1}P \P. Then μ is finite if and only

if ∑_{n=1}^{∞} d(q, π_{pq}(f^{n}z)) is convergent.

Proof. Let P′_{n} = {x ∉ P : f^{i}x ∈ P, 1 ≤ i ≤ n}. We have μP = ∑_{n=1}^{∞} μ̄P′_{n}

since P = ⋃_{n=1}^{∞} f^{n}P′_{n}. μP′_{n} is proportional to the length l(γ_{n}^{u}) of any unstable

curve γ_{n}^{u} ⊂ P′_{n} that crosses P′_{n}, because (3.1) also holds for the density function of the conditional measures of μ. Now we use the fact that l(γ_{n}^{u}) is proportional to d(q, π_{pq}(f^{n}z)). □

Corollary 3.5. Suppose f : M → M is an almost Anosov diffeomorphism with only one indifferent fixed point p. Let ϕ : M → ℝ be a continuous function.

If f admits an SRB measure μ, then for Leb-a.e. x ∈ M,

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

If f admits an infinite SRB measure μ, then for Leb-a.e. x ∈ M,

lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} ϕ(f^{i}x) = ϕ(p).

The second part of the corollary is true because the measure of any neighborhood of p is infinite, while the measure of its complement is finite.

As we mentioned before that if f admits an infinite SRB measure, then it is statistically deterministic and topologically chaotic. Here is another feature.

Corollary 3.6. If f has an infinite SRB measure, then the forward Lyapunov exponents are 0 for Leb-a.e. x ∈ M.

Proof. Take ϕ(x) = log |Df_{x}|_{E_{x}^{u}}| and log |Df_{x}|_{E_{x}^{s}}| in the above corollary. Continuity of ϕ follows from continuity of E_{x}^{u} and E_{x}^{s} for x ≠ p and the fact Df_{x} → Df_{p} = id as x → p. □

It seems that whether a system has an SRB measure depends on whether expansion is stronger than contraction near the indifferent fixed point. Here we discuss some special cases. Assume that Df_{p} has eigenvalues \overline{λ} ≥ \underline{λ} > 0.

Fact 3.7. If \overline{λ} > 1, then f has an SRB measure.

This is because the system is uniformly expanding along unstable directions. So distortion for Df|_{E^{u}} is bounded. Therefore the same “push-forward” method gives an probability invariant measure.

If \underline{λ} < 1 besides \overline{λ} > 1, then the system is an Anosov system.

Theorem 3.8 ([HY]). If \overline{λ} = 1, \underline{λ} < 1, then f has an infinite SRB measure.

Proof. By (2.3) and Fact 2.1, d(p, f^{−n}z) is of order n^{−1/2k} ∀z ∈ W_{loc}^{u}(p). Then the result follows from Proposition 3.4 and the fact that the stable foliation is C^{1}. □

If Df_{p} = id, then we need look at the third order terms of the Taylor expression of the map at p. We only have some sufficient conditions to guarantee that both cases in Theorem 3.3 do occur. Assume that we can choose a suitable coordinate system such that f can be written as

f ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}x + x(ax^{2} + rxy + by^{2} + O((x^{2} + y^{2})^{3/2}) ) \\ y − y(cx^{2} + sxy + dy^{2} + O((x^{2} + y^{2})^{3/2}) ) \end{matrix} ) .

Theorem 3.9 ([H1]). With the above notations, i) f has an SRB measure if b > 2d, r = 0 = s, and ad > bc; ii) f has an infinite SRB measure if 2b < d and rs ≠ 0.

Proof. Part i) follows from Proposition 2.6 and 3.4. For Part ii), we can proof that d(q, π_{pq}(f^{−n}z)) is bounded below by O(n^{−1/2}). □

The conditions in the first part say a/c > b/d > 2. So expansion is stronger. For the second part, we can see that at least there is a cone on the plane in which contraction is stronger.

If the only eigenvalue of Df_{p} is 1 and Df_{p} ≠ id, then near p, f has the form

f(x, y) = (x + ay + r(x, y), y + s(x, y)),

where a ≠ 0, and r and s consist of higher order terms of x and y. Theorem 3.10 ([CE]). If the coefficient of x^{2} in r is zero and coefficient of x^{3} in s is nonzero, then f has an SRB measure.

3.2. Absolutely Continuous Invariant Measures. It is well-known that piecewise smooth uniformly expanding systems on the unit interval have absolutely continuous invariant measures. (A folklore theorem, also see e.g. [KS], [W] for higher dimensional results.) For almost expanding maps on the unit interval, it is proved by G. Pianigiani [Pi] that absolutely continuous invariant measures exist, which may be finite or σ-finite.

The proof in [Pi] uses the first return maps. Here we give an alternative one. This method can be easily adopted to a more general setting.

Theorem 3.11. Let f : I → I be a piecewise smooth expanding map with an indifferent fixed point 0. Assume that near 0, f satisfies (2.1). Then f has an absolutely continuous invariant measure μ, and μ is finite if 0 < γ < 1, σ-finite if γ ≥ 1.

Proof. Take ϕ(x) = − log f′(x). Define the Perron-Frobenius operator by

L_{ϕ}g(x) = ∑_{fx_{1}=x} e^{ϕ(x_{1})}g(x_{1}).

Since the fixed point of L_{ϕ} is unbounded near 0, we take a Banach space B = C^{0}(0, 1] but with the norm ∥g∥ = sup{x^{2γJ}g(x) : x ∈ I}. Then we construct a convex set

G = {g ∈ B : g > 0, g(1) = 1, ∃H_{0} s.t. x^{γ}g(x) ≤ H_{0} ∀x ∈ I_{0},

g(y) ≤ g(x)(1 + Jx^{−1}d(x, y) ∀x, y ∈ P_{n}}. (3.2)

Note that as x → 0, x^{2γJ}g(x) → 0 uniformly for all g ∈ G. So G is compact in B.

Let L̃_{ϕ}g = L_{ϕ}g/L_{ϕ}g(1). Clearly, L̃_{ϕ} is continuous. It can be proved that L̃_{ϕ}G ⊂ G. So by Schauder-Tychonoff fixed point theorem, L̃_{ϕ} has a fixed point h ∈ G. Hence, L_{ϕ}h = h. Using h as the density function, we get an invariant measure μ. By the corollary below, h is integrable if and only if γ < 1. □

The following result tells that on P_{n}, h(x) increases like γnh̃(0), where h̃(x) =

∑_{x̂_{1}∈f^{−1}x\I_{0}} \frac{h(x̂_{1})}{f′(x̂_{1})}. (Recall Fact 2.1.)

Corollary 3.12. lim_{x→0} x^{γ}h(x) = h̃(0).

Proof. Observe that L_{ϕ}h = h implies that for x ∈ I_{0}, if x_{1} = f^{−1}x ∩ I_{0}, then

x_{1}^{γ}h(x_{1}) ∼ x^{γ}h(x)\frac{x_{1}^{γ}}{x^{γ}} · f′(x_{1}) − h̃(0)x_{1}^{γ}f′(x_{1}),

where ∼ means that the difference between the two sides is of order higher than x^{γ}. Since f′(x) ∼ 1 + (1 + γ)x^{γ} and x^{γ}/x_{1}^{γ} ∼ 1 + γx_{1}^{γ}, we have

x_{1}^{γ}h(x_{1}) ∼ x^{γ}h(x)(1 + x_{1}^{γ}) − h̃(0)x_{1}^{γ}.

So if there is c > 0 such that x^{γ}h(x) ≥ h̃(0)(1 + c) for a small x, then

x_{1}^{γ}h(x_{1}) ≥ h̃(0)(1 + c + 0.5cx_{1}^{γ})

because (1 + c)(1 + x_{1}^{γ}) − x_{1}^{γ} = 1 + c + cx_{1}^{γ}. Inductively, x_{n}^{γ}h(x_{n}) ≥ h(0)(1 + c + 0.5c ∑_{i=1}^{n} x_{i}^{γ}). Since ∑_{i=1}^{n} x_{i}^{γ} diverges, it contradicts the fact that x^{γ}h(x) ≤ H_{0} for all x ∈ I_{0}. So the upper limit is bounded by h̃(0). The lower limit can be proved similarly. □

By checking the proof carefully, we can actually get the following.

Theorem 3.13. Let f : Λ → Λ be a piecewise smooth almost expanding map with an indifferent fixed point. Suppose ϕ = − log | det Df_{x}| satisfies the following.

i) There is J_{ϕ} > 0 such that S_{n}ϕ(y_{n}) − S_{n}ϕ(x_{n}) ≤ J_{ϕ}x^{−1}d(x, y) ∀x, y ∈

P_{n}, x_{n} ∈ f^{−n}x, y_{n} ∈ f^{−n}y ∩ R_{n}(x_{n}).
ii) On P_{n}, ϕ(x) ≤ −\frac{β + 1}{n} for some β > 0.

Then f has an absolutely continuous invariant measure μ. Further, μ is finite if

β > 1, and σ-finite if moreover ϕ(x) ≥ −\frac{β′ + 1}{n} on P_{n} for some 0 < β′ ≤ 1.

Theorem 3.11 says that we can use γ to determine finiteness of absolutely continuous invariant measures of almost expanding systems on intervals. It is unclear what the higher dimensional versions are. However, in some simple cases, we can get similar results, which can be regarded as an application of Theorem 3.13.

Assume that an almost expanding map f : I^{m} → I^{m} has the form

f(x) = x(1 + |x|^{γ}) + o(x^{1+γ})

as x near 0. That is, f is a map that expands at the same rate in all directions at 0, plus a higher order perturbation. Then with the same methods, we have:

Fact 3.14. f has an absolutely continuous invariant measure μ, which is finite if 0 < γ < m, and σ-finite if γ ≥ m.

Proof. Check that on P_{n}, ϕ ∼ −\frac{1 + m/γ}{n}. This also implies the first condition in the above theorem. □

However, in higher dimensional spaces, ϕ may varies, say, from −(1 + β_{1})/n to −(1 + β_{2})/n, β_{1} ≠ β_{2}, on P_{n}. If so, the first condition in Theorem 3.13 usually fails. In this case we need modify the proof. Here is an example.

Fact 3.15. Suppose f : I^{2} → I^{2} is an almost expanding map and near 0,

f ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) = ( \begin{matrix}x_{1} + x_{1}^{1+γ_{1}} + o(x_{1}^{1+γ_{1}}) \\ x_{2} + x_{2}^{1+γ_{2}} + o(x_{2}^{1+γ_{2}}) \end{matrix} ) .

Then f has an absolutely continuous invariant measure μ, which is finite if γ_{1}^{−1} + γ_{2}^{−1} > 1, and σ-finite if γ_{1}^{−1} + γ_{2}^{−1} ≤ 1.

Proof. We can partition each P_{n} into countable many smaller pieces, such that on each of them the map has bounded distortion. To make the set G compact, we use x_{1}^{2γ_{1}J}x_{2}^{2γ_{2}J}g(x_{1}, x_{2}) to define the norm in the space B. To compute integral of return time, we compute the sum of the area of O_{n}, which are rectangles of size approximately (γ_{1}n)^{−1/γ_{1}} × (γ_{2}n)^{−1/γ_{2}} by Fact 2.1. □

If an almost expanding system has an absolutely continuous invariant measure μ, and it contains only one indifferent fixed point p, then a statement similar in Corollary 3.5 can be made. That is, for Lebesgue almost every point, the time average is equal to ∫ ϕdμ if μ is finite, and is equal to ϕ(p) if μ is σ-finite.

If f has two indifferent fixed points that cause σ-finite measure, then from the case of one dimensional systems, the limits for time average may not exist.

Theorem 3.16 ([I]). Suppose f is a piecewise smooth expanding map on the interval with indifferent fixed points p and q such that

|f(x) − x| = a|x − p|^{1+α} + o(|x − p|^{1+α}), x → p;
|f(x) − x| = b|x − q|^{1+β} + o(|x − q|^{1+β}), x → q.

Then for any small neighborhoods U and V of p and q respectively, Leb-a.e. x,

lim_{n→∞} \frac{∑_{i=0}^{n} 1_{U}(f^{i}(x))}{∑_{i=0}^{n} 1_{V} (f^{i}(x))} = \{ \begin{matrix}0 & if β > α ≥ 1, \\ c > 0 & if β = α = 1, \\ doesn’t exist & otherwise. \end{matrix}

3.3. Measures of Full Hausdorff Dimensions. A measure ν on a bounded subset Λ in ℝ^{n} is called an measure of full Hausdorff dimension if HD(ν) = HD(Λ), where HD(Λ) is the Hausdorff dimension of Λ, and

HD(ν) = inf{HD(Λ_{0}) : Λ_{0} ⊂ Λ, ν(Λ\Λ_{0}) = 0}.

If Λ is an invariant set of a conformal expanding map, then f has an invariant probability measure μ such that HD(μ) = HD(Λ). However, for an invariant set of an almost expanding map, the measure of full Hausdorff dimension may be σ-finite.

Theorem 3.17 ([U]). Let 0 ∈ Λ ⊂ I be an invariant set of a piecewise smooth almost expanding map f with δ = HD(Λ). Assume that near 0, f satisfies (2.1). Then f has an invariant measure μ of full Hausdorff dimension, and μ is finite if δ > 2γ/(1 + γ); μ is σ-finite if δ ≤ 2γ/(1 + γ).

Proof. Take ϕ(x) = −δ log f′(x). On P_{n}, we have ϕ(x) ∼ δ(1 + γ^{−1})/n. Then we use a similar method for Theorem 3.13. The role of the Lebesgue measure is replaced by a conformal measure ν which is a fixed point of the dual operator of L_{ϕ} on the space C^{0}(Λ). □

By works of M. Urbański et al, Λ differ from hyperbolic Cantor sets in many properties. (See [U] for details and related references.) HD(Λ) is always bounded below by γ/(1 + γ), while the Hausdorff dimensions of hyperbolic Cantor sets can be arbitrary small. The topological pressure P (f|_{Λ}, −t log f′) is nonnegative. (This is because near 0, e^{S_{n}ϕ(x_{n})} decreases in a polynomial rate.) The pressure is positive if t < HD(Λ), and 0 if t ≥ HD(Λ). That is, HD(Λ) is the smallest solution of P (f|_{Λ}, −t log f′) = 0. For hyperbolic Cantor sets, P (f|_{Λ}, −t log f′), as a function of t, is strictly decreasing, and HD(Λ) is the unique solution of the equation.

3.4. Weak Gibbs States and Equilibrium States. We cannot expect the measures obtained in Section 3.1- 3.3 to be Gibbs States for the potential functions. This is because at an indifferent fixed point p, we may have e^{S_{n}ϕ(p)} = 1 for any n, and μO_{n} → 0 in a polynomial rate. Therefore there is no uniform bound for the ratios between μR_{n}(p) and exp{−nP + S_{n}ϕ(p)}. Instead, we consider week Gibbs states.

The notion of weak Gibbs state was introduced by R. Dobrushin, and then studied by C. Maes et al (see e.g. [MRM] for related references). The definition we use below is from [Yu2]. Recall that R_{n}(x) was defined in Section 1.

Definition 3.18. A Borel probability measure μ is called a weak Gibbs state for a function ϕ with constant P if there is a sequence {C_{n}} of positive number with lim_{n→∞}(1/n) log C_{n} = 0 such that for μ-a.e. x,

(3.3)  C_{n}^{−1} ≤ \frac{μR_{n}(x)}{exp\{−nP + S_{n}ϕ(x)\}} ≤ C_{n}.

It is proved in [Yu2] and [MRTMV] that absolute continuous invariant probability measures for systems discussed in Theorem 3.11 are week Gibbs states. Actually, all probability measures we discussed in previous sections are weak Gibbs states.

A weak Gibbs state of ϕ is an equilibrium state of the function, and the constant P is the topological pressure. In fact, since μ is a probability measure, (3.3) gives

C_{n}^{−1} ≤ e^{−nP} ∑ e^{S_{n}ϕ(x_{B})} ≤ C_{n},

where the sum is taken over all elements B ∈ R_{n} and x_{B} is any point in B. By the definition, we know that P is the topological pressure. Also, by (3.3),

\frac{1}{n}S_{n}ϕ(x) − \frac{1}{n} log C_{n} ≤ P + \frac{1}{n} log μR_{n}(x) ≤ \frac{1}{n}S_{n}ϕ(x) + \frac{1}{n} log C_{n}.

Since lim_{n→∞}(1/n)S_{n}ϕ(x) = ∫ ϕdμ and lim_{n→∞} −(1/n) log R_{n}(x) = h_{μ}(σ) for μ-a.e.x by the Birkhoff ergodic theorem and the Shannon-McMillan-Breiman theorem respectively, the inequality implies P = ∫ ϕdμ + h_{μ}(σ).

It is clear that equilibrium states are not unique for almost hyperbolic systems since obviously the point mass δ_{p} is another one.

4. Convergence Rate

We discuss decay rates of correlations and convergence rates of functions under the Perron-Frobenius operators.

4.1. Decay of Correlations. The correlation of two functions g, ĝ : M → ℝ is

ρ_{g,ĝ}(n) = ∫ g · (ĝ ∘ f^{n})dμ − ∫ gdμ ∫ ĝdμ.

If a system (f, μ) is mixing, then for any g, ĝ ∈ L^{2}(μ), lim_{n→0} ρ_{g,ĝ}(n) = 0. So we can

think that the speed of the convergence reflects the speed of mixing. (We refer the paper by V. Baladi in this volume for general information and related references about decay of correlations.)

One of important characters for almost hyperbolic systems is that they can have polynomial decay of correlations.

Definition 4.1. We say that (f, μ) has polynomial decay of correlations with degree α for functions in F, if for all g, ĝ ∈ F, ∃C = C(g, ĝ) > 0 s.t.

|ρ_{g,ĝ}(n)| ≤ Cn^{−α} ∀n ≥ 1.

Polynomial decay of correlations was first proved in 1993 by Mori ([M]) and by Lambert-Siboni-Vaienti ([LaSV]) for piecewise linear maps on the unit interval with an indifferent fixed point (Takahashi model). For piecewise smooth almost expanding maps, the result was proved by Young [Y2], Liverani-Saussol-Vaienti ([LiSV]), Pollicott-Yuri ([PY]), and the author [H2], all with different approaches. The lower bounds of the decay rates for the systems are also polynomial. (See Theorem 4.3.) For invertible systems, polynomial decay is proved for some almost Anosov systems on surfaces (Theorem 4.5).

Applying the results in ([Y2]) to an almost hyperbolic system, we get that decay rates of correlations are determined by mO_{n}, where m is a reference measure. Here we introduce slightly different conditions, which is given in terms of invariant measures ([H3]).

Let f be an almost expanding map preserving an invariant measure μ. Suppose μ ∘ f << μ. Put

ψ(x) = − log \frac{dμ ∘ f}{dμ} (x).

If μ is an absolutely continuous measure with density function h, then it is easy to check that ψ(x) = − log(| det Df_{x}| · h(fx)/h(x)).

Let

L_{ψ}g = ∑_{x_{1}∈f^{−1}x} e^{ψ(x_{1})}g(x_{1})

be the Perron-Frobenius operator. Since μ is an f-invariant measure, L_{ψ}c = c for any constant c and μ(L_{ψ}g) = μ(g) for any integrable function g.

Theorem 4.2 ([H3]). Suppose ψ satisfies the following:

i) there is J_{ψ} > 0, θ ∈ (0, 1) such that S_{n}ψ(y_{n}) − S_{n}ψ(x_{n}) ≤ J_{ψ}d(x, y)^{θ}

∀x, y ∈ P_{n}, where x_{n} ∈ f^{−n}x, y_{n} ∈ f^{−n}y ∩ R_{n}(x_{n});
ii) on P_{n}, ψ(x) ≤ −\frac{β}{n} + O(\frac{1}{n^{1+r}} ), r > 0, for some β > 1, or
ii’) ∀k > 0, ∃C_{ψ} > 0 such that e^{S_{n}ψ(x_{n})} ≤ C_{ψ}(n + k)^{−β}, ∀x ∈ P_{k}.

Then (f, μ) has polynomial decay of correlations with degree β − 1.

Clearly, condition ii) implies ii’). The latter implies μO_{n} = O(n^{−β}).

Proof of Theorem 4.2: It follows directly from Theorem 4.6 in the next subsection, because ρ_{g,ĝ}(n) = ∫ (ĝ ∘ f^{n}) · (g − μ(g))dμ = ∫ ĝ · (L_{ψ}^{n}g − μ(g))dμ, where we use the fact that L_{ψ} is the dual of the operator F : L^{2}(μ) → L^{2}(μ) defined by F g(x) = g(fx). □

For one dimensional systems, we have the following, where β = 1/α.

Theorem 4.3. Let f be a piecewise smooth expanding map on the interval with an indifferent fixed point 0. Assume that f satisfies (2.1) with 0 < γ < 1. Then

i) ([Y2], [H2]) for all Lipschitz functions g, ĝ, there exists C = C(g, ĝ) s.t.

|ρ_{g,ĝ}(n)| ≤ Cn^{−(β−1)} ∀n ≥ 1;

ii) ([H2]) there exist C^{∞} functions g, ĝ and a constant C′ > 0 s.t.

|ρ_{g,ĝ}(n)| ≥ C′n^{−(β−1)} ∀n ≥ 1.

Proof of Part i): Distortion estimates for ψ follow from that for ϕ and the fact that h ∈ G, defined in (3.2). For the second condition, we consider x ∈ P_{n}.

We know ϕ(x) ∼ −\frac{β + 1}{n} by Fact 2.1. By Corollary 3.12, \frac{h(fx)}{h(x)} ∼ \frac{n − 1}{n}. So

ψ(x) ∼ −\frac{β + 1}{n} + \frac{1}{n} = −\frac{β}{n}. □

The proof for Part ii) needs different method. For lower bounds in general case, we have:

Conjecture 4.4. Under the circumstances of Theorem 4.2, if moreover ψ(x) ≥

−\frac{β′}{n} + O(\frac{1}{n^{1+r}} ), r > 0, for some β′ ≥ β, then the lower bounds of decay rates of

the system are of order O(n^{−(β′−1)}).

The systems discussed in Fact 3.14 and Theorem 3.17 are also have polynomial decay of correlations with the degrees mγ^{−1} − 1 and δ(1 + γ^{−1}) − 1 respectively, whenever the measures we discussed are finite.

Pollicott and Yuri ([PY]) studied decay rate of correlations of the inhomogeneous Diophantine approximation transformation, defined by

f(x, y) = (\frac{1}{x} − [\frac{1 − y}{x} ] + [−\frac{y}{x}], −[−\frac{y}{x}] − \frac{y}{x}),

where 0 ≤ y ≤ 1, −1 ≤ x ≤ −y+1, and [x] is the integer part of x. This map has two indifferent periodic points (1, 0) and (−1, 1), and admits an absolutely continuous measure with density 1/(2 log 2)(1−x^{2}). The rate of decay of correlations is proved to have bounds O(n^{−1+ϵ}) ∀ϵ > 0. If we look at the behavior of the map near the periodic orbit, the bounds should be of oeder O(n^{−1}).

For invertible systems, we have:

Theorem 4.5 ([H5]). Let f : M → M be an almost Anosov diffeomorphism on the surface that satisfies conditions in Theorem 3.9. i). Then f has polynomial decay of correlations with degree b/2d − 1.

Proof. (f, μ) can be reduced to an almost expanding map (f̃, μ̃) by identify points on each stable curve in every rectangle of a Markov partition. Then we can check that the resulting measure μ̃ satisfies the conditions in Theorem 4.2. In Particular, if x ∈ P_{n}, then e^{ψ(x)} is proportional to μP_{n−1}/μP_{n} = μP′_{n−1}/μP′_{n} by the definition of ψ. By measuring the length of unstable leaves of P′_{n} (Proposition 2.6), we know

that it is bounded by (\frac{n − 1}{n})^{β}, β = b/2d. So we get ψ(x) ≤ −β/n, and therefore

(f̃, μ̃) has decay rate bounded by O(n^{−(β−1)}). Then we pass the results from (f̃, μ̃) to (f, μ). □

4.2. Convergence to Density Functions. In this section we study the rate of convergence of L_{ψ}^{n}g → μ(g). This is the rate that a system converges to its equilibrium, and the rate that determines the rate of decay of correlations.

Theorem 4.6. Under the circumstance of Theorem 4.2, for any Lipschitz function g, there is a constant C > 0 such that for all n > 0,

∫ |L_{ψ}^{n}g − μ(g)|dμ ≤ \frac{C}{n^{β−1}} ;

and for any closed subset E with p ∉ E, there is C_{1} > 0 such that for all n > 0,

|L_{ψ}^{n}g(x) − μ(g)| ≤ \frac{C_{1}}{n^{β−1}} ∀x ∈ E.

Recall the definition of L_{ψ}. Denote g_{n} = L_{ψ}^{n}g. For x ∈ P_{n}, n > 0, write

L_{ψ}g(x) = e^{ψ(x_{1})}g(x_{1}) + (1 − e^{ψ(x_{1})})ḡ(x_{1}),
where ḡ(x) = ∑_{x̃∈f^{−1}(fx)∩P_{0}} e^{ψ(x̃)}g(x̃)/ ∑_{x̃∈f^{−1}(fx)∩P_{0}} e^{ψ(x̃)}, the average of g with the

weight e^{ψ} at all preimages of fx except for x itself. It is easy to see that

L_{ψ}^{n}g(x) = g(x_{n})e^{S_{n}ψ(x_{n})} + ∑_{j=1}^{n} ḡ_{n−j}(x_{j})(1 − e^{ψ(x_{j})})e^{S_{j−1}ψ(x_{j−1})}. (4.1)

Proof of Theorem 4.6: Note that O_{k+1} is the complement of Q_{k}. We proof the following for some large k by induction on n:

(A_{n}): |g_{n}(x) − μ(g)| ≤ \frac{C}{(n + 1)^{β−1}} for any x ∈ Q_{k};

(B_{n}): ∫_{O_{k+1}} |g_{n}(x) − μ(g)|dμ ≤ \frac{C}{(n + 1)^{β−1}} .

“(A_{j}), j = 0, ⋯, n − 1, ⇒ (B_{n})”: Apply (4.1) with the function g(x) − μ(g). Since ∫_{O_{k+1}} e^{S_{n}ψ(x_{n})}dμ = μO_{k+n+1} has the order O(n^{−(β−1)}) by the condition ii’) in Theorem 4.2, so is ∫_{O_{k+1}} e^{S_{n}ψ(x_{n})}|g_{n}(x) − μ(g)|dμ.

For the second term, we note that ∫_{O_{k+1}}(1 − e^{ψ(x_{j})})e^{S_{j−1}ψ(x_{j−1})} = μO_{k+j} − μO_{k+j+1} = μP_{k+j} has the order O((k + j)^{−β}), and g_{n−j}(x) − μ(g) has the order O((n − j)^{−(β−1)}) by the induction hypotheses (A_{n−j}). So the sum is bounded by C′n^{−(β−1)} for some small C′ if k is large enough.

“(B_{n}) ⇒ (A_{n})”: By (B_{n}), ∫_{Q_{k}}(g_{n}(x) − μ(g))dμ ≤ C(n + 1)^{−(β−1)}. By Proposition 4.7 below, g_{n}(y) − g_{n}(x) is of order O(n^{−β}) for any x, y ∈ Q_{k}. So (A_{n}) must be true. □

By the proof we can see that the first term in (4.1), and therefore μO_{n}, determines the rate of convergence.

Proposition 4.7. Let g > 0 be a Lipschitz function, and let k > 0. Then there is A > 0 such that for any x, y ∈ Q_{k},

\frac{g_{n}(y)}{g_{n}(x)} ≤ 1 + \frac{A}{(n + 1)^{β}} ∀n > 0.

This is the main step towards estimates of the speed of convergence L_{ψ}^{n}g → μ(g). We use projective metric (Hilbert metric) to prove the proposition, since under this metric L_{ψ} is a contracting operator. Let

C_{J}(S) = {g ∈ C^{0}(S) : g(x) > 0 ∀x ∈ S, g(y) ≤ g(x)e^{Jd(x,y)} ∀x, y ∈ S}

be a cone in C^{0}(S), where C^{0}(S) is the space of continuous functions on a metric space S. The projective metric of two functions g, g̃ ∈ C_{J}(S) is given by

Δ_{J}(g, g̃) = log \frac{a(g, g̃)}{b(g, g̃)},

where

a(g, g̃) = inf{a : ag̃ − g ∈ C_{J}(S)}, b(g, g̃) = sup{b : g − bg̃ ∈ C_{J}(S)}.

We will always take g̃ ≡ 1. So we denote Θ_{J}(g) = Δ_{J}(g, 1) = log(a_{g}/b_{g}), where

a_{g} = inf\{a : \frac{a − g(y)}{a − g(x)} ≤ e^{Jd(x,y)}\}, b_{g} = sup\{b : \frac{g(y) − b}{g(x) − b} ≤ e^{Jd(x,y)}\}.

It is easy to see that to measure how much a function g differs from a constant function, we can use Θ_{J}(g), or g(y) − g(x), or 1 − g(y)/g(x).

Proof of Proposition 4.7: Suppose k is large. Take l large enough. Recall

Q_{k} = ⋃_{i=0}^{n} P_{i}. Define Q_{k,l} = ⋃_{i=k}^{l} P_{i}. We prove the following for all n ≥ 0:

(C_{n}): Θ_{2J}(g_{n}|_{Q_{k+l}}) ≤ \frac{A}{(n + 1)^{β}};

(D_{n}): Θ_{J}(g_{n}|_{Q_{k,l}}) ≤ \frac{c}{k^{1/γ}} · \frac{A}{(n + 1)^{β}};

(E_{n}): Θ_{J}(g_{n}|_{Q_{k}}) ≤ (1 − \frac{c}{k^{1/γ}} ) · \frac{A}{(n + 1)^{β}}.

Clearly, (D_{n}) and (E_{n}) imply (C_{n}). Since L_{ψ} is contracting in the projective metric, we can get that (C_{n}) implies (E_{n+l}), if l is large enough. To complete the induction process, we only need the fact that (C_{j}), j = 0, ⋯ n − 1, imply (D_{n}). It is enough to have g_{n}(y) − g_{n}(x) ≤ cA(n + 1)^{−β} for some small c > 0.

Recall (4.1). For the first term, since e^{S_{n}ψ(x_{n})} has the order O(n^{β}) by the assumption in Theorem4.2 ii’), e^{S_{n}ψ(y_{n})}g(y_{n}) − e^{S_{n}ψ(x_{n})}g(x_{n}) has the same order.

For the second term, we note that d(x_{j}, y_{j}) has the order j^{−α} for some α > 1 since f is expanding in a polynomial rate near the indifferent fixed point. Then we use (C_{n−j}) to obtain that ḡ_{n−j}(y_{j}) − ḡ_{n−j}(x_{j}) ≤ C(n − j + 1)^{−β}d(x_{j}, y_{j}) ≤ C_{1}(n − j + 1)^{−β}j^{−α} ≤ C_{2}(n + 1)^{−β}. Since (1 − e^{ψ(x_{j})})e^{S_{j−1}ψ(x_{j−1})}, j = 1, ⋯, n, has the sum 1 − e^{S_{n}ψ(x_{n})}, it can be treated by the same way as the first term. □

If μ is a σ-finite measure, we cannot expect L_{ψ}^{n}g → μ(g) since L_{ψ}^{n}g(x) → 0 for Lebesgue almost every x whenever |μ(g)| < ∞. However, if we multiply the left side by suitable factors, we may still have convergence to the density function.

Theorem 4.8 ([CF]). Suppose f is an almost expanding map on [0, 1] that maps both (0, 1/2) and (1/2, 1) to (0, 1). Assume that near 0, fx = x + ax^{2} + O(x^{3}). Then for any real function g bounded away from 0 and ∞,

lim_{n→∞} \frac{A_{n}}{n} ∑_{i=0}^{n−1} L_{ϕ}^{i}g = h

uniformly on any compact subset E ⊂ (0, 1] for some h with L_{ϕ}h = h, where A_{n} is of order log n.

By Proposition 4.7, if 0 < γ < 1, then restricted to Q_{k}, the difference between L_{ψ}^{n}g and a constant is of order O(n^{−β}). Based on this observation, we make the following conjecture for one dimensional case.

Conjecture 4.9. Under the circumstance of Theorem 4.3 with γ ≥ 1, for any Lipschitz function g, there exist a_{n} = O(n^{−β}), β = 1/γ, such that

lim_{n→∞} A_{n}L_{ϕ}^{n}g = h

uniformly on any compact subset E ⊂ (0, 1] for some h with L_{ϕ}h = h, where

A_{n} = ∏_{i=0}^{n−1}(1 + a_{i}). Further, the convergence rate is of order n^{−β}.

Since {A_{n}} is convergent if γ ∈ (0, 1), the conjecture is consistent with Theorem 4.6.

References

[B] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lecture Notes in Math. Vol. 470 (Springer-Verlag, Berlin-New York, 1975).

[BY] M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measure for certain Hénon maps, Invent. Math. 112 (1993), 541–576.

[BV] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193.

[CE] E. Catsigeras and H. Enrich, SRB measures of certain almost hyperbolic diffeomorphisms with a tangency, Preprint.

[CF] P. Collet and P. Ferrero, Some limit ratio theorem related to a real endomorphism in case of neutral fixed point, Ann. Inst. Henri Poicaré 52 (1990), 283–301.

[G] M. Gerber, Conditional stability and real analytic pseudo-Anosov maps, Mem. Amer.

Math. Soc. 54 (1985), no. 321.

[GK] M. Gerber and A. Katok Smooth models of Thurston’s pseudo-Anosov maps, Ann. Sci. École Norm. Sup. (4), 15 (1982), 173–204.

[H1] H. Hu, Conditions for the Existence of SBR Measures for “Almost Anosov” Diffeomorphisms, Trans. Amer. Math. Soc. 352 (2000) 2331–2367.

[H2] H. Hu, Decay of Correlations for Piecewise Smooth Maps with Indifferent Fixed Points, Preprint.

[H3] H. Hu, Projective metrics and polynomial decay of correlations, Preprint.

[H4] H. Hu, Equilibrium states with polynomial decay of correlations, Preprint.

[H5] H. Hu, Decay of correlations for some almost Anosov diffeomorphisms, Preprint.

[HPK] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math. Vol. 583 (Springer-Verlag, Berlin-New York, 1977).

[HY] H. Hu and L.-S. Young, Nonexistence of SBR measures for some diffeomorphisms that

are “almost Anosov”, Ergod. Th. & Dynam. Sys. 15 (1995), 67–76.

[I] T. Inoue, Ratio ergodic theorems for maps with indifferent fixed points, Ergod. Th. &

Dynam. Sys. 17 (1997), 625–642.

[JN] M. Jakobson and S. Newhouse, On the structure of non-hyperbolic attractors, in Dynamical systems and chaos, Vol. 1, ed. N. Aoki, etc. (World Sci. Publishing, River Edge, 1972), pp 103–111.

[K] A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2) 110 (1979), 529– 547.

[KS] K. Krzyzewski and W. Szlenk, On invariant measure for expanding differentiable mappings, Studia Math. 33 (1969), 83–92.

[LS] F. Ledrappier and J.-M. Strelcyn, Estimation from below in Pesin’s entropy formula,

Ergod. Th. & Dynam. Sys. 2 (1982 ), 203–219.

[LaSV] A. Lambert, S. Siboni and S. Vaienti, Statistical properties of a nonuniformly hyperbolic map of the interval, J. Statist. Phys. 72 (1993), 1305–1330.

[LiSV] C. Liverani, B.Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergod.

Th. & Dynam. Sys. 19 (1999), 671–68.

[Lo] J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations 38 (1980), 192–209.

[M] M. Mori, On the intermittency of a piecewise linear map, Tokyo J. Math. 16 (1986),

411–428.

[MRM] C. Maes, F. Redig and A. Moffaert, Almost Gibbsian versus weakly Gibbsian measures,

Stoch. Proc. Appl. 79 (1999), 1-15.

[MRTMV] C. Maes, F. Redig, F.Takens, A. Moffaert and E. Verbitski , Intermittency and weak

Gibbs states, Preprint.

[N] S. Naoya, Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures, Trans. Amer. Math. Soc. 351 (1999), 3373–3385.

[P] Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties, Ergod. Th. & Dynam. Sys. 12 (1992), 123-151.

[Pi] G. Pianigiani, First return map and invariant measures in case of neutral fixed point, Israel J. Math. 35 (1980), 32–48.

[PY] M. Pollicott and M. Yuri, Statistical Properties of maps with indifferent periodic

points, Preprint.

[S] Ya. G. Sinai, Gibbs measure in ergodic theory, Russian Math. Surveys 27 (1972), 21–69.

[U] M. Urbański, Parabolic Cantor sets , Fund. Math. 151 (1996), 241-277.

[W] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121–153.

[Y1] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147 (1998), 585–650.

[Y2] L.-S. Young, Recurrence time and rate of mixing, Israel J. Math. 110 (1999), 153–188.

[Yu1] M. Yuri, Invariant measures for certain multi-dimensional maps, Nonlinearity 7 (1994), 1093–1124.

[Yu2] M. Yuri, Weak Gibbs measures for certain nonhyperbolic systems, Ergod. Th. & Dynam. Sys. (to appear).

Department of Mathematics, Pennsylvania State University, University Park, PA 16802

E-mail address: hu@math.psu.edu

Current address: Department of Mathematics, University of Southern California, Los Angeles, CA 90089

E-mail address: huyihu@math.usc.edu