INFIMUM OF THE METRIC ENTROPY OF HYPERBOLIC ATTRACTORS WITH RESPECT TO THE SRB MEASURE

HUYI HU, MIAOHUA JIANG, AND YUNPING JIANG

Abstract. Let M^{n} be a compact C^{∞} Riemannian manifold of dimension n ≥ 2. Let Diff^{r}(M^{n}) be the space of all C^{r} diffeomorphisms of M^{n}, where 1 < r ≤ ∞. For a C^{r} diffeomorphism f in Diff^{r}(M^{n}) with a hyperbolic attractor Λ_{f} on which f is topologically transitive, let U(f) be the C^{1} open set of Diff^{r}(M^{n}) such that each element in U(f) can be connected to f by finitely many C^{1} structural stability balls in Diff^{r}(M^{n}). Then by the structural stability, any element g in U(f) has a hyperbolic attractor Λ_{g} and g|Λ_{g} is topologically conjugate to f|Λ_{f}. Therefore, the topological entropy h(g|Λ_{g}) is a constant function when it is restricted to U(f). However, the metric entropy h_{μ}(g) with respect to the SRB measure μ = μ_{g} can vary. We prove that the infimum of the metric entropy h_{μ}(g) on U(f) is zero.

0. Introduction

Let M^{n} be a compact C^{∞} Riemannian manifold of dimension n ≥ 2, and Diff^{r}(M^{n}) be the space of all C^{r}-diffeomorphisms of M^{n}, 1 < r ≤ ∞. Suppose f ∈ Diff^{r}(M^{n}) has a hyperbolic attractor Λ_{f}. Consider an open set U(f) ⊂ Diff^{r}(M^{n}) in C^{1}-topology consisting of C^{r} diffeomorphisms g that has a hyperbolic attractor Λ_{g} topologically conjugate to Λ_{f}. (The precise definition of U(f) is given in Section 1.) Since the hyperbolic attractors of any two elements in U(f) are topologically conjugate, the topological entropy h(g|Λ_{g}) is a constant function restricted to U(f). On the other hand, every g ∈ U(f) has an SRB measure μ_{g} on Λ_{g}. The metric entropy h_{μ_{g}}(g) can vary in U(f). It has been shown that the dependence of μ_{g} on the map g is smooth when the maps involved have a higher degree of smoothness (see [Ru] and references therein). In this article, we prove that the infimum of the metric entropy h_{μ_{g}}(g) over U(f) is zero.

In the special case of Anosov systems, if we denote by 𝒜^{r}(M^{n}) the set of all C^{r} Anosov diffeomorphisms of M^{n}, and let U(f) be the connected component of 𝒜^{r}(M^{n}) in C^{r} topology that contains f, then by structure stability, all elements g in U(f) are topologically conjugate and therefore the topological entropy h(g) is constant over U(f), but the metric entropy h_{μ_{g}}(g) over U(f) with respect to the SRB measures μ_{g} can be arbitrarily small.

This result is interesting in several aspects. First, the topological entropy tells the global complexity of a dynamical system. If a dynamical system has a positive topological entropy, it can be thought as a chaotic dynamical system. However, the topological entropy can not tell the level of complexity of a chaotic dynamical system. As a macroscopic quantity, the metric entropy is the standard value that measures the level of complexity of a chaotic dynamical system. Our result says that, given a hyperbolic attractor, there is no barrier to reduce its metric entropy along a C^{1} homotopic path to a number as small as one wishes, while preserving the uniform hyperbolicity and the topological entropy. Second, our construction of the homotopy gives an example where the connection between a global quantity such as metric entropy and local perturbations can be concretely described. Third, the result leads to many interesting questions one may ask about the nature of the variation of metric entropy within the open neighborhood U(f). For example, is it true that the maximal value of the metric entropy h_{μ_{g}}(g) on U(f) is the value of the topological entropy h(f)? Is there a way to perturb the diffeomorphism f in a direction so that its metric entropy h_{μ_{g}}(g) either decreases or increases monotonically? Are there any local extrema of the metric entropy h_{μ_{g}}(g) in U(f)?

To prove this result, the idea of perturbing a map at a fixed point in the direction of unstable manifold is quite natural. Since μ is an SRB measure, we have h_{μ}(f) = ∫ log | det Df_{x}|_{E_{x}^{u}}|dμ. So if we reduce the expanding rate near a fixed point p, a typical orbit will spend more time near p where | det Df_{x}|_{E_{x}^{u}}| is close to 1. But the approach by considering such an orbit is difficult because we do not know whether a typical orbit will remain typical after perturbation.

In order to get control of the SRB measure for perturbed maps, we need distortion estimates of the unstable Jacobian along orbits independent of the ever-decreasing expansion rate at the fixed point. We apply a technique used in studying almost Anosov systems ([HY], [Hu1]). However, in this paper, we deal with a family of diffeomorphisms {f_{t} : 0 ≤ t ≤ 1} whose expansion decreases to 1 near a fixed point p and for the limiting map f_{0}, the expanding rate at p is 1. We obtain that the stable foliation is absolute continuous and the Jacobian of the holonomy maps are uniformly bounded for all f_{t}, and the distortion estimates of the unstable Jacobian are also uniformly bounded for all f_{t} as long as the initial points of the backward orbits are away from p.

Our construction also gives that every diffeomorphism with hyperbolic attractor is homotopic to a diffeomorphism that has an almost hyperbolic attractor, and has an infinite SRB measure on it.

The rest of the article is divided into 5 sections. The statement of results is given in the next section. In Section 2 we describe in detail how the perturbation is constructed. We show that any map f ∈ Diff^{r}(M^{n}) with a hyperbolic attractor can be perturbed successively in an appropriate way within U(f) such that the perturbed map has a desired normal form in a neighborhood of its fixed point. In Section 4, we prove the uniform boundedness of the distortion of the unstable Jacobian for all perturbed maps. In Section 5, we give proofs of our theorems.

1. Statement of Results

For the definition of standard terms such as uniform hyperbolicity, topological conjugacy, Lyapunov exponents, topological and metric entropies, we refer readers to the book [KH].

Suppose that μ is an invariant measure for f ∈ Diff^{r}(M^{n}) that has positive Lyapunov exponents at almost every point x. Then f has a unstable manifold W^{u}(x) at such x. A measurable partition ξ of M^{n} is said to be subordinate to unstable manifolds if for μ-a.e x, ξ(x) ⊂ W^{u}(x) and ξ(x) contains an open neighborhood of x in W^{u}(x). Let {μ_{x}^{ξ}} denote a canonical system of conditional measures of μ with respect to ξ, that is, for every measurable set B ⊂ M^{n}, x → μ_{x}^{ξ}(B) is measurable and

ν(B) = ∫_{X} ν_{x}^{ξ}(B)dν(x).

(For a reference, see e.g. [Ro].) We say that a measure μ on Λ_{f} has absolutely continuous conditional measures on unstable manifolds, if for every measurable partition ξ subordinate to unstable manifolds, μ_{x}^{ξ} is absolutely continuous with respect to m_{x}^{u} for μ-a.e. x ∈ Λ_{f}, where m_{x}^{u} denotes the Lebesgue measure induced on W^{u}(x) (see [LS] for more details). Now we give a definition for SRB measure which can be found in [BY] (see also e.g. [HY], [Hu1]).

Definition 1. An f-invariant Borel probability measure μ on M is called an Sinai-Ruelle-Bowen measure or 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 mani-

folds.

It is well known that if μ is an SRB measure on a hyperbolic attractor of f, then for any continuous function ψ on M, we have

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

for Lebesgue almost every x in the basin of the attractor (see [S]). Also, an SRB measure μ is the only invariant measure satisfying the variational principle

(1.1)  h_{μ}(f) = ∫_{Λ} log | det Df_{x}|_{E_{x}^{u}}|dμ,

where h_{μ}(f) is the metric entropy of f with respect to μ ([Bo]). This formula also follows from the entropy formula and the fact that the righthand side of (1.1) is equal to the integral of the sum of the positive Lyapunov exponents.

A map f ∈ Diff^{r}(M^{n}) is said to possess a hyperbolic attractor Λ_{f} if there is an open set V ⊃ Λ_{f} such that f is hyperbolic on Λ_{f} and ∩_{n=1}^{∞}f^{n}(V) = Λ_{f}.

Suppose f ∈ Diff^{r}(M^{n}) has a hyperbolic attractor Λ_{f}. By the structural stability, there is a sufficiently small ϵ-neighborhood of f with respect to C^{1} topology (called a C^{1} structural stability ball),

B_{ϵ}^{1}(f) = {g ∈ Diff^{r}(M^{n}) | ∥f − g∥_{1} < ϵ},

where ∥ · ∥_{1} means the C^{1} norm, such that any g ∈ B_{ϵ}^{1}(f) has a hyperbolic attractor Λ_{g} and g|Λ_{g} is topologically conjugate to f|Λ_{f}. We say that a map g ∈ Diff^{r}(M^{n}) can be connected with f if there are finitely many neighborhoods {B_{ϵ_{i}}^{1}(f_{i})}_{i=0}^{n} such that f_{0} = g and f_{n} = f and f_{i} ∈ B_{ϵ_{i+1}}^{1}(f_{i+1}), 0 ≤ i ≤ n − 1. Let U(f) be the collection of such diffeomorphisms g in Diff^{r}(M^{n}) which can be connected with f in the above sense. It is clear that U(f) is an open set of Diff^{r}(M^{n}) with respect to the C^{1}-topology. If restricted to the hyperbolic attractors, any two maps in U(f) are topologically conjugated by a Hölder continuous homeomorphism. Hence they have the same topological entropy.

For g ∈ U(f), let μ_{g} be its SRB measure on Λ_{g}.

For convenience we assume that f is topologically transitive on Λ_{f}. Hence, all g ∈ U(f) are topologically transitive on Λ_{g}, and the SRB measrues are unique. If otherwise we can use spectral decomposition ([Bo]) and consider a topologically transitive component instead.

Theorem A. Suppose f ∈ Diff^{r}(M^{n}) has a hyperbolic attractor Λ_{f} on which f is topologically transitive. Then there is a C^{1} path

H = {f_{t} ∈ U(f) | 0 < t ≤ 1}

such that f_{1} = f and

(1.2)  lim_{t→0^{+}} h_{μ_{t}}(f_{t}) = 0,

where μ_{t} denotes the unique SRB measure of f_{t} on the hyperbolic attractor Λ_{f_{t}}.

Remark 1.1. If f is an Anosov diffeomorphism, then U(f) is a connected component of the subspace 𝒜^{r}(M^{n}) of all Anosov diffeomorphisms with respect to C^{1}-topology in Diff^{r}(M^{n}). The theorem says that on any connected component of 𝒜^{r}(M^{n}), the infimum of h_{μ}(f) is zero.

Corollary 1.2. As t → 0, μ_{t} → δ_{O(p)} in the weak^{∗} topology, and

h_{μ}(f_{t}) → 0 = h_{δ_{O(p)}}(f_{0}),

where p is a fixed or periodic point, δ_{O(p)} is the invariant measure supported on the orbit O(p) of p.

Our proof of the theorem also gives that the limiting diffeomorphism f_{0} has an almost hyperbolic attractor Λ_{f_{0}}, on which f_{0} admits an infinite SRB measure. A closed f-invariant subset Λ ⊂ M is called an almost hyperbolic set if it is hyperbolic everywhere except for a finite set S. We refer [Hu2] for precise definition.

The following definition of an infinite SRB measure for an almost hyperbolic attractor can be found in [Hu1]. Here for a subset Γ ⊂ M, we denote by f_{Γ} the first return map on Γ, and by μ_{Γ} the normalization of μ|Γ as μ(Γ) < ∞.

Definition 2. An f-invariant Borel measure μ on M is called an infinite SBR measure, if μ(M) = ∞ and for any open set V ⊃ S,

i) μ(M\V) < ∞, ii) (f_{M\V} , μ_{M\V}) has positive Lyapunov exponents almost everywhere, and μ_{M\V} has absolutely continuous conditional measures on unstable manifolds of f.

We say that maps f and g are C^{r}-homotopy in a subset ℱ ⊂ Diff^{r}(M^{n}) if there is a continuous map H : [0, 1] × M^{n} → M^{n} such that H(0, ·) = f, H(1, ·) = g, H(t) = H(t, ·) ∈ ℱ ∀t ∈ [0, 1], and H(·) : [0, 1] → ℱ is continuous with respect to C^{r} topology for ℱ. We also denote by \overline{U(f)} the closure of U(f) in C^{r} topology.

Theorem B. Any diffeomorphism f in Diff^{r}(M^{n}) that has a hyperbolic attractor is C^{r}-homotopic to a diffeomorphism f_{0} in \overline{U(f)} that has an almost hyperbolic attractor on which f_{0} admits an infinite SRB measure.

Remark 1.3. In the case of Anosov systems, the theorem implies that any Anosov diffeomorphism f is C^{r}-homotopic to an almost Anosov diffeomorphism in \overline{𝒜(M^{n})} that admits an infinite SRB measure.

In [HY], it is proved that there is an almost Anosov diffeomorphism on the two dimensional torus that has an infinite SRB measure. This theorem generalizes the result to almost hyperbolic attractors and to higher dimensional manifolds.

Remark 1.4. By the same arguments as in [HY] we can prove that f_{0} does not admit SRB measures, and lim_{n→∞} ∑_{i=0}^{n−1} δ_{f^{i}x} = δ_{O(p)} for Lebesgue almost every point in the basin of the attractor.

2. Construction

Suppose f ∈ Diff^{r}(M^{n}) has a hyperbolic attractor Λ_{f} on which f is topologically transitive. Without loss of generality, we will assume that f has a fixed point p. Otherwise, we can consider a periodic orbit and deform the maps in the same way near every point in the orbit. We will construct a family

\{f_{t} ∈ \overline{U(f)} | 0 ≤ t ≤ 1\}

of maps having hyperbolic attractors Λ_{f_{t}} on which f_{t} are topologically transitive. The family satisfies the following main properties:

(1) The fixed point p is preserved for the family and maps in the family are obtained by perturbing f in a small neighborhood of p.

(2) The contracting rates along the stable directions for the maps in the family are bounded above by a constant strictly less than 1, while the expanding rates along the unstable directions for the maps in the family in a small neighborhood of p can be arbitrarily close to 1 as t → 0^{+}.

2.1. Linearizing the map near p. We first show the following lemma.

Lemma 2.1. For the given map f as above, there is g ∈ Diff^{r}(M^{n}) having the same fixed point p such that the following properties hold:

(1) g ∈ U(f) is C^{1} close to f, and C^{r}-homotopy to f in U(f).

(2) g is identical to f outside an ϵ_{1}−neighborhood of p.

(3) There is a coordinate system η : ℝ^{n} → M^{n} near p such that η(0) = p and η^{−1} ∘ g ∘ η is a linear map in an ϵ_{0} (< ϵ_{1})− open ball centered at 0 ∈ ℝ^{n} of the form

(2.1)  L (\begin{matrix}x^{u} \\ x^{s} \end{matrix}) = (\begin{matrix}Ax^{u} \\ Bx^{s} \end{matrix}) ,

where x = (x^{u}, x^{s}) are coordinates provided by the unstable and stable subspaces of Df at p, A is an expanding linear map, and B is an contracting linear map.

Let n^{u} and n^{s} be the dimensions of the unstable and stable subspaces of Df at p. We may assume that the coordinate system provided by the unstable and stable subspaces are just Euclidean spaces ℝ^{n^{u}} and ℝ^{n^{s}} with ℝ^{n^{u}} ⊗ ℝ^{n^{s}} = ℝ^{n}.

Proof. We identify p with the origin 0 of ℝ^{n}. There exists an ϵ_{1}-neighborhood of p in M^{n} such that f is C^{1} close to its linear approximation Df(0) in this ϵ_{1}-neighborhood. Identify this ϵ_{1}-neighborhood with an open ball B_{ϵ_{1}}(0) of radius ϵ_{1} centered at 0 in ℝ^{n}.

In the coordinate system given by the unstable and stable subspaces ℝ^{n^{u}} and ℝ^{n^{s}}, the linear operator Df(0) takes the form

Df(0) (\begin{matrix}x^{u} \\ x^{s} \end{matrix}) = (\begin{matrix}Ax^{u} \\ Bx^{s} \end{matrix}) ,

Take a smaller open ball B_{ϵ_{0}}(0) of radius 0 < ϵ_{0} < ϵ_{1} centered at 0. Extend the map (f − Df(0))|_{B_{ϵ_{0}}(0)} to a C^{r} map f_{h} on B_{ϵ_{1}}(0) so that ∥f_{h}∥ + ∥Df_{h}∥ is small. This can be done as long as the ratio ϵ_{0}/ϵ_{1} is small enough. Let

D(ϵ_{0}) = sup_{x∈B_{ϵ_{0}}(0)}{∥f(x) − Df(0)x∥ + ∥Df(x) − Df(0)x∥}.

We can require that

∥f_{h}∥ + ∥Df_{h}∥ ≤ 2D(ϵ_{0}).

Clearly,

lim_{ϵ_{0}→0} D(ϵ_{0}) = 0.

Let τ(x) be a smooth map such that it is the identity inside B_{ϵ_{0}}(0) and it maps every point to zero outside of B_{ϵ_{1}}(0). Define

g = \begin{cases}f & if x ∉ B_{ϵ_{1}}(0); \\ f − τ ∘ f_{h} & if x ∈ B_{ϵ_{1}}(0). \end{cases}

The map g is linear in B_{ϵ_{0}}(0). The C^{1}-distance between g and f is bounded by the ∥f_{h}∥ + ∥Dτ∥∥Df_{h}∥. Note that by selecting a small ratio ϵ_{0}/ϵ_{1}, the map τ can be made as smooth as we wish with derivatives uniformly bounded. So, the C^{1}-distance between g and f can be is less than any given number. It implies that g ∈ Diff^{r}(M^{n}) is in a C^{1}-neighborhood of f. Thus, g has a hyperbolic attractor Λ_{g} on which g is topologically transitive. Clearly, g is C^{r}-homotopic to f in U(f). This completes the proof. □

Note that, other than using Df(0)x, the linear approximation, we can use other nonlinear choices near 0. The only condition to be satisfied is that these choices are C^{r} and sufficiently C^{1}-close to f in an ϵ_{1}-neighborhood of 0.

Corollary 2.2. The linear map in (2.1) near the origin 0 can be changed to any C^{r} map which is C^{1} close to f in a neighborhood of 0, in particular, a map in the form of

(2.2)  x = (\begin{matrix}x^{u} \\ x^{s} \end{matrix}) → (\begin{matrix}(A + C(x^{s}))x^{u} \\ Bx^{s} \end{matrix}) ,

where C(x^{s}) are a linear maps for each x^{s}, depending on x^{s} in C^{r} and satisfying C(0) = 0.

Now we further perturb the map g inside the small neighborhood B_{ϵ_{1}}(0). The perturbation preserves the direct product structure form (2.1) near the point 0. We need to perturb the map along a homotopy path in U(f) so that the matrices A and B become diagonal matrices first. In the unstable direction, we need to further reduce the expanding rates to a number close to 1. In the stable direction, the contracting rate will be a constant. The perturbation is no longer in a small C^{1} neighborhood of f. But we will show that the perturbed maps are still uniformly hyperbolic and homotopic to f in U(f).

Lemma 2.3. Let g be the map obtained in Lemma 2.1. Assume that a C^{r} diffeomorphism g_{1} ∈ Diff^{r}(M^{n}) has the following properties:

(1) g_{1}(x) = f(x) for x ∉ B_{ϵ_{0}}(0) and g_{1}(0) = 0.

(2) g_{1}(x) preserves the direct product structure ℝ^{n^{u}} and ℝ^{n^{s}} for x ∈ B_{ϵ_{0}}(0), i.e., g_{1}(x) takes the form

g_{1} (\begin{matrix}x^{u} \\ x^{s} \end{matrix}) = (\begin{matrix}Ã(x^{u}) \\ B̃(x^{s}) \end{matrix}) if x = (\begin{matrix}x^{u} \\ x^{s} \end{matrix}) ∈ B_{ϵ_{0}}(0),

where Ã(x^{u}) ∈ ℝ^{n^{u}} and B̃(x^{s}) ∈ ℝ^{n^{s}}.

(3) Ã(x^{u}) is expanding on ℝ^{n^{u}} and B̃(x^{s}) contracting on ℝ^{n^{s}}, both uni-

formly. Then, when ϵ_{0} is sufficiently small, g_{1} has a hyperbolic attractor Λ_{g}.

A direct corollary of this lemma is that when Ã(x^{u}) and B̃(x^{s}) are homotopic to the linear maps Ax^{u} and Bx^{s} in Diff^{r}(M^{n}), then g_{1} is in U(f).

Proof. Since we assume that f has a hyperbolic attractor Λ_{f} on which f is topologically transitive, so is g in Lemma 2.1. Let Λ_{g} be the corresponding hyperbolic attractor of g. Take an ϵ-neighborhood O_{ϵ}(Λ_{g}) such that the stable and unstable subspaces are extended to the entire neighborhood of the hyperbolic set Λ_{g} [KH]. We may also assume that ϵ_{0} ≤ ϵ, thus the entire B_{ϵ_{0}}(0) is contained in O_{ϵ}(Λ_{g}). Take an appropriate coordinate system (almost Lyapunov metric) on the neighborhood O_{ϵ}(Λ_{g}) such that the stable and unstable subspaces are nearly orthogonal. At each point x inside the set B_{ϵ_{0}}(0), the unstable and stable subspaces E_{x}^{u} and E_{x}^{s} for Dg(x) are within Cϵ_{0}-distance (Grassmannian distance) of ℝ^{n^{u}} and ℝ^{n^{s}}, respectively, for some constant C. We assume that for some λ < 1 < μ, we have

∥Dg(x)v∥ ≥ μ∥v∥, v ∈ E_{x}^{u}; ∥Dg(x)w∥ ≤ λ∥w∥, w ∈ E_{x}^{s}.

To show that g_{1} is uniformly hyperbolic, we first prove that both Dg_{1} and D^{−1}g_{1} admit an invariant cone field and they induce contracting operators on these cone fields. As a consequence, we obtain exponential splitting along any orbit of g_{1} inside the open neighborhood O_{ϵ}(Λ_{g}). We then show the expanding and contracting rates along the invariant subspaces of the exponential splitting are uniform. In the process of the proof, we may need to further decrease the radius ϵ_{0} for B_{ϵ_{0}}(0). But such modification will not affect the validity of the arguments since it depends only on the map g.

The cone field C_{x}^{u} along the unstable subspaces is defined as follows.

C_{x}^{u} = \begin{cases}\{(v, w) ∈ T_{x}M^{n} : v ∈ E_{x}^{u}, w ∈ E_{x}^{s}, ∥w∥ ≤ α∥v∥ for x ∉ B_{ϵ_{0}}(0)\}; \\ \{(v, w) ∈ T_{x}M^{n} : v ∈ ℝ^{n^{u}}, w ∈ ℝ^{n^{s}}, ∥w∥ ≤ α∥v∥ for x ∈ B_{ϵ_{0}}(0)\}, \end{cases}

where 0 < α < 1. The cone field C_{x}^{s} along the stable subspaces is defined in a similar way. Note that this cone field is not continuous. But this should not affect the Hölder continuity of stable and unstable subspaces because of invariance.

We just need to show that the derivative operator is contracting on this cone field, since invariance follows the contraction. Given a point x, if both x and g_{1}(x) are inside or outside of B_{ϵ_{0}}(0), the contraction follows automatically since E^{u} and E^{s} are invariant under Dg_{1} when it is restricted to the outside of B_{ϵ_{0}}(0) and since ℝ^{n^{u}} and ℝ^{n^{s}} are invariant under Dg_{1} when it is restricted to the inside of B_{ϵ_{0}}(0). So, we just need to verify two situations: x ∈ B_{ϵ_{0}}(0) but g_{1}(x) ∉ B_{ϵ_{0}}(0); and g_{1}(x) ∈ B_{ϵ_{0}}(0) but x ∉ B_{ϵ_{0}}(0).

Now we consider the case x ∈ B_{ϵ_{0}}(0) but g_{1}(x) ∉ B_{ϵ_{0}}(0).

Take (v, w) ∈ C_{x}^{u}, v ∈ E_{x}^{u}, w ∈ E_{x}^{s}. Then Dg_{1}(x)(v, w) = (DÃv, DB̃w) is in ℝ^{n^{u}} ⊕ ℝ^{n^{s}} coordinate system. In the E^{u} ⊕ E^{s} coordinate system, we have

(P_{11}DÃv + P_{12}DB̃w, P_{21}DÃv + P_{22}DB̃w) ∈ E^{u} ⊕ E^{s},

where P_{ij} denotes the coordinate change matrices and DÃ, DB̃ are derivative operators of Ã(x^{u}) and B̃(x^{s}). Since p is a fixed point of g and g_{1} is C^{0}-close to g, ∥g_{1}(x)∥ ≤ 2ϵ_{0}. So, we have ∥P_{11}∥, ∥P_{22}∥ ≥ (1 − ϵ) and ∥P_{12}∥, ∥P_{21}∥ ≤ ϵ where ϵ = Cϵ_{0} for some constant C. Estimating the norms, we have

∥P_{11}DÃv + P_{12}DB̃w∥ ≥ ((1 − ϵ)μ − αϵλ̃)∥v∥;
∥P_{21}DÃv + P_{22}DB̃w∥ ≤ (ϵμ̃ + αλ(1 − ϵ))∥v∥,

where μ̃, λ̃ denote the maximal expanding rate and minimal contracting rate of Ã and B̃, respectively. We have

(2.3) ∥P_{21}DÃv + P_{22}DB̃w∥ ≤ α · \frac{λ(1 − ϵ) + ϵμ̃/α}{(1 − ϵ)μ − αϵλ̃} · ∥P_{11}DÃv + P_{12}DB̃w∥.

If ϵ_{0} is sufficiently small, we have

\frac{λ(1 − ϵ) + ϵμ̃/α}{(1 − ϵ)μ − αϵλ̃} < 1.

The proof for the other case is completely parallel. Thus we proved invariance of the cone field around the stable subspaces.

We now prove that g_{1} is uniform hyperbolic. We only consider the unstable direction. The proof for stable direction is the same. Since p is a fixed point, we can take 0 < ϵ_{2} ≤ ϵ_{0} such that B_{ϵ_{2}}(0) ∪ g_{1}B_{ϵ_{2}}(0) ∪ g_{1}^{−1}B_{ϵ_{2}}(0) ⊂ B_{ϵ_{o}}(0). So for any x, either both x and g_{1}(x) are outside of B_{ϵ_{2}}(0), or both x and g_{1}(x) are inside of B_{ϵ_{0}}(0). Note that the estimates obtained in (2.3) remain valid if we replace ϵ_{0} by a smaller number ϵ_{2}. Thus, if both x and g_{1}(x) are outside of B_{ϵ_{2}}(0), taking (v, w) ∈ C_{x}^{u}, ∥w∥ = α̃∥v∥ for some 0 ≤ α̃ ≤ α < 1, and using the invariant splitting of g, we have

∥Dg_{1}(x)(v, w)∥ = ∥(Dg_{1}(x)v, Dg_{1}(q)w)∥
= max(∥Dg_{1}(x)v∥, ∥Dg_{1}(x)w∥) = ∥Dg_{1}(x)v∥ ≥ μ∥v∥,

where we use an equivalent Finsler metric in estimating the expanding rates. If both x and g_{1}(x) are inside of B_{ϵ_{0}}(0), using the ℝ^{n^{u}} and ℝ^{n^{s}} coordinates, we have the same estimates in Finsler metric. We thus conclude that g_{1} is uniformly hyperbolic. □

Corollary 2.2 and Lemma 2.3 give us plenty of freedom to perturb the map f inside a fixed neighborhood B_{ϵ_{0}}(0). In order to obtain bounded distortion along unstable manifold, we will need to use Lemmas 2.1 and 2.3 to obtain a map g_{1} ∈ U(f) in the form of (x^{u}, x^{s}) → (Ã(x^{u}), B̃(x^{s})). Then, we apply Corollary 2.2 to obtain another map in U(f) that has the form (x^{u}, x^{s}) → (Ã(x^{u}) + C(x^{s})x^{u}, B̃(x^{s})) near the fixed point p.

2.2. Constructing the map f_{t}. We now construct a family

H = {f_{t} ∈ U(f) | 0 ≤ t ≤ 1}

such that the infimum of metric entropies of maps in this family with respect to their SRB measures is zero.

Recall that n^{s} and n^{u} are dimensions of ℝ^{n^{s}} and ℝ^{n^{u}}, respectively. Take an even integer m such that n^{u} < m ≤ n^{u} + 2, and denote β = 1/m. Hence, βn^{u} < 1. Take a family of smooth increasing functions ϕ_{t} : [0, 1/2) → ℝ^{+} that satisfies the following conditions.

(1) ϕ_{t}(r) = (1 − r^{m})^{−β} for r ≥ t,

(2) ϕ_{t}(r) = (1 − (t/2)^{m})^{−β} > 1 is a constant for 0 ≤ r ≤ t/2,

(3) ϕ′′_{t}(r) is bounded uniformly in t and nonnegative when \frac{t}{2} ≤ r ≤ t. Note that if r = 1/n^{β} ∈ [t, 1/2), then ϕ′(r) > 0 and

\frac{1}{n^{β}} ϕ_{t}(\frac{1}{n^{β}}) = \frac{1}{(n − 1)^{β}} . (2.4)

When t = 0, rϕ_{0}(r) is essentially a special Manneville-Pomeau map.

Take two open neighborhoods Ω_{0} ⊂ Ω_{1} such that p ∈ Ω_{0} ⊂ \overline{Ω}_{0} ⊂ Ω_{1}. Denote κ^{s} = ∥Df(0)|_{E_{0}^{s}}∥, the contracting rate at 0 in the stable direction. Define a family of diffeomorphisms f_{t} such that

(1) for x ∈ Ω_{0},

(2.5)  f_{t} (\begin{matrix}x^{u} \\ x^{s} \end{matrix}) = (\begin{matrix}[ϕ_{t}(|x^{u}|) + |x^{s}|^{2}]x^{u} \\ κ^{s}x^{s} \end{matrix}) , if x = (\begin{matrix}x^{u} \\ x^{s} \end{matrix});

(2) for x ∉ Ω_{1}, f_{t}(x) = f(x);

(3) f_{t} is uniformly hyperbolic for all t > 0 and the contracting rate is independent of t;

(4) the expanding rate outside of Ω_{0} is bounded below by a constant μ > 1 with μ independent of t.

We also require that the second derivative of f_{t} is uniformly bounded for all t and x ∈ M^{n}. For simplicity, we use | · | to denote both the absolute value of a number and the Euclidean norm of a vector. The existence of f_{t} is guaranteed by Lemma 2.1, Corollary 2.2, and Lemma 2.3 when t is sufficiently small. We note that f_{t} is C^{∞} on Ω_{0} since m is an even number. It is clear that f_{t} is C^{r}-homotopic to f for all 0 < t ≤ 1. Hence, f_{t} has an hyperbolic attractor Λ_{f_{t}}, and therefore f_{t} ∈ U(f) for all 0 < t ≤ 1 by the statement after Lemma 2.3.

3. Preliminaries

Being uniformly hyperbolic, the maps constructed in the previous section have many special properties. We list some of them with only sketches of proofs since details can be found in many books such as [KH, HPS]. Let E_{x}^{u}(f_{t}) and E_{x}^{s}(f_{t}) be the stable and unstable subspaces with respect to f_{t}.

Lemma 3.1. The maps x → {E_{x}^{u}(f_{t})} and x → {E_{x}^{s}(f_{t})} are Hölder continuous and the Hölder exponents and constants can be chosen in a way independent of 0 ≤ t ≤ 1. More precisely, there exist γ > 0, H > 0 such that for all t > 0, x, x′ ∈ M^{n},

d(E_{x}^{u}(f_{t}), E_{x′}^{u}(f_{t})), d(E_{x}^{s}(f_{t}), E_{x′}^{s}(f_{t})) ≤ Hd(x, x′)^{γ},

where the distance between subspaces is the Grassmannian distance.

Proof. Note that ∥Df_{t}(x)|_{E_{x}^{u}(f_{t})}∥ ≥ 1 and ∥Df_{t}(x)|_{E_{x}^{s}(f_{t})}∥ ≤ λ^{s} < 1 for all x ∈ Λ_{f_{t}} and t ≥ 0. The conclusions follow from the fact that the Hölder exponent and constant depend only on the Lipschitz constant of the map f_{t} and the gap between the expansion and contraction rates. See [HPS]. □

For ε > 0, we denote

E_{x,f_{t}}^{u}(ε) = {v ∈ E_{x}^{u}(f_{t}) : |v| ≤ ε} and E_{x,f_{t}}^{s}(ε) = {v ∈ E_{x}^{s}(f_{t}) : |v| ≤ ε}

and

E_{x,f_{t}}(ε) = E_{x,f_{t}}^{u}(ε) × E_{x,f_{t}}^{s}(ε).

Proposition 3.2. For each t ≥ 0, there exist two continuous foliations ℱ^{u}(f_{t}) and ℱ^{s}(f_{t}) on Λ_{f_{t}} tangent to E^{u}(f_{t}) and E^{s}(f_{t}) respectively for which the following statements hold.

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

ℱ^{s}(x, f_{t}) = W^{s}(x, f_{t})

= {x′ ∈ Λ_{f_{t}} : ∃C = C′_{x}, s.t. d(f_{t}^{n}(x), f_{t}^{n}(x′)) ≤ C(κ_{t}^{s})^{n} ∀n ≥ 0},

where κ_{t}^{s} denotes the contracting rate of f_{t} on the stable manifold.

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

ℱ^{u}(x, f_{t}) = W^{s}(x, f_{t}) = {x′ ∈ Λ_{f_{t}} : lim_{n→∞} d(f_{t}^{−n}(x), f_{t}^{−n}(x′)) = 0}.

(3) There exist constants δ > 0 and D > 0 such that for all t ≥ 0, x ∈ Λ_{f_{t}}, if ℱ_{δ}^{u}(x, f_{t}) is the component of ℱ^{u}(x, f_{t}) ∩ exp_{x} E_{x,f_{t}}(δ) containing x, then exp_{x}^{−1} ℱ_{δ}^{u}(x, f_{t}) is the graph of a function

ϕ_{x,f_{t}}^{u} : E_{x,f_{t}}^{u}(δ) → E_{x,f_{t}}^{s}(δ)

with ϕ_{x,f_{t}}^{u}(0) = 0 and ∥ϕ_{x,f_{t}}^{u}∥_{C^{r}} ≤ D. The analogous statement holds for ℱ_{δ}^{s}(x, f_{t}).

Proof. There results follow from Theorem 5.5 and Theorem 5A.1 in [HPS]. □

For convenience we will write W^{u}(x, f_{t}) = ℱ^{u}(x, f_{t}), W_{δ}^{u}(x, f_{t}) = ℱ_{δ}^{u}(x, f_{t}), etc. and refer to W^{u}(x, f_{t}) and W_{δ}^{u}(x, f_{t}) as the “unstable manifold” and “local unstable manifold” respectively at x.

Further, for simplicity, we use E^{u}, E^{s}, W^{u} and W^{s} to stand for invariant subspaces and sub-manifolds, when doing so will not cause much confusion.

For x′ ∈ W^{s}(x, f_{t}), let d^{s}(x, x′) denote the distance between x and x′ measured along W^{s}(x, f_{t}) with the induced metric. For x′′ ∈ W^{u}(x, f_{t}), the distance d^{u}(x, x′′) is defined similarly.

Lemma 3.3. There is J^{s} > 0 such that for any x ∈ Λ_{f_{t}}, x′ ∈ W^{s}(x, f_{t}),

\frac{| det Df_{t}^{n}(x)|_{E_{x}^{u}(f_{t})}|}{| det Df_{t}^{n}(x′)|_{E_{x′}^{u}(f_{t})}|} ≤ J^{s}(d^{s}(x, x′))^{γ}, ∀t ≥ 0.

Proof. It follows from the standard arguments and the fact that f_{t} is uniformly contracting along the stable manifold and E_{x}^{u}(f_{t}) is Hölder continuous. □

One of the important ingredients in the proof of the main theorem is that the W^{s}-foliation for f_{t} is absolute continuous with a uniformly bounded Jacobian.

Let Δ_{1} and Δ_{2} be two W^{u} leaves for f_{t}. A holonomy map θ : Δ_{1} → Δ_{2} is defined by sliding along the W^{s} leaves for f_{t}, i.e. for x ∈ Δ_{1}, θ(x) ∈ Δ_{2} ∩ W^{s}(x, f_{t}). We have the following proposition.

Proposition 3.4. Given D_{1} > 0, there exists J_{1}^{s} > 0 such that for every (Δ_{1}, Δ_{2}; θ) with d^{s}(x, θ(x)) < D_{1} ∀x ∈ Δ_{1}, for every x′ ∈ Δ_{1} and ε > 0 with B^{u}(x′, ε) ⊂ Δ_{1},

m^{u}(B^{u}(x′, ε)) ≤ J_{1}^{s}m^{u}(θB^{u}(x′, ε)).

Proof. Let 𝒟 be any small disk in Δ_{1}. We will argue that m^{u}(𝒟) ≈ m^{u}(θ𝒟), where m^{u} denotes the Lebesgue measure on W^{u}-sub-manifolds for f_{t}, and “≈” means “up to a constant”.

Let

κ_{−}^{s} = κ_{−}^{s}(f_{t}) = max{∥Df_{t}^{−1}(x)|_{E_{x}^{s}(f_{t})}∥^{−1} : x ∈ Λ_{f_{t}}},

that is, κ_{−}^{s} is the minimal norm of Df_{t} restricted to the stable bundle E^{s}(f_{t}).

By taking a sufficiently large iterate of f_{t}, we may assume that f_{t}^{n}(𝒟) and f_{t}^{n}(θ𝒟) are close sufficiently so that the “diameters” of f_{t}^{n}(𝒟) and f_{t}^{n}(θ𝒟) are much larger than the distance d^{s}(y, θy) for any y ∈ f_{t}^{n}(𝒟). Here we also use θ to denote the holonomy map from f_{t}^{n}(Δ_{1}) to f_{t}^{n}(Δ_{2}).

Take a finite cover {B^{u}(y_{i}, r)}_{i=1}^{k} of f_{t}^{n}𝒟 consisting balls of radius r, where r = 3(κ_{−}^{s})^{n}, such that for any y′ ∈ f_{t}^{n}(𝒟), there are at most C_{1} such balls covering this point, where C_{1} only depends on the dimension of W^{u}.

Since d^{s}(y′, θy′) ≤ (κ_{−}^{s})^{n} for any y′ ∈ f_{t}^{n}(𝒟), we know that θB^{u}(y_{i}, r) contains a ball of radius at least (κ_{−}^{s})^{n} and is contained in a ball of radius at most 5(κ_{−}^{s})^{n}. Therefore, we have

(3.1)  m^{u}(B^{u}(y_{i}, r)) ≈ m^{u}(θB^{u}(y_{i}, r)).

By Lemma 3.3, we have

(3.2) |det Df_{t}^{n}(x′)|_{E_{x′}^{u}(f_{t})}| ≈ |det Df_{t}^{n}(θx′)|_{E_{θx′}^{u}(f_{t})}| ∀x′ ∈ f_{t}^{−n}B^{u}(y_{i}, r).

Using the fact that the C^{2} norm of f_{t} is uniformly bounded, W^{u}(x, f_{t}) are C^{r} leaves, and f_{t} is expanding on W^{u}(x, f_{t}), we get for all x′_{1}, x′_{2} ∈ f_{t}^{−n}B^{u}(y_{i}, r),

(3.3) \frac{|det Df_{t}^{n}(x′_{1})|_{E_{x′_{1}}^{u}(f_{t})}|}{|det Df_{t}^{n}(x′_{2})|_{E_{x′_{2}}^{u}(f_{t})}|} ≈ ∏_{j=0}^{n−1}(1 ± const · d^{u}(f_{t}^{j}x′_{1}, f_{t}^{j}x′_{2}) )

≤(1 ± const · diam B^{u}(y_{i}, r))^{n} ≈ (1 ± const · D_{1}(3κ_{−}^{s})^{n})^{n} ≈ const.

Combining (3.1)-(3.3), we get

(3.4)  m^{u}(f_{t}^{−n}B^{u}(y_{i}, r)) ≈ m^{u}(f_{t}^{−n}(θB^{u}(y_{i}, r))).

Let S be the union of f_{t}^{−n}B^{u}(y_{i}, r) that belong to 𝒟. Since

diam(f_{t}^{−n}B^{u}(y_{i}, r)), diam(θ(f_{t}^{−n}B^{u}(y_{i}, r))) → 0 as n → ∞,

it follows that

m^{u}(S) ≈ m^{u}(𝒟) and m^{u}(θS) ≈ m^{u}(θ𝒟).

Also recall that each point in 𝒟 belongs to at most C_{1} sets of the form f_{t}^{−n}B^{u}(y_{i}, r), then (3.4) implies m^{u}(𝒟) ≈ m^{u}(θ𝒟). □

4. Distortion Estimates

We now estimate the distortion of the Jacobian along the unstable direction. We use the ℝ^{n^{u}} ⊕ ℝ^{n^{s}}-coordinate near the fixed point 0 and use Euclidean metric for convenience. Let

S = S_{δ} = {x = (x^{u}, x^{s}) ∈ Ω_{1} : |x^{u}|, |x^{s}| ≤ δ}.

Denote S^{+} = \overline{f_{t}(S) \ S}. It is clear that S^{+} is independent of t for all small t.

Let δ^{+} = δϕ_{t}(δ), where ϕ_{t} is given in Subsection 2.2. It is the outer radius of S^{+}.

We denote y = (y^{u}, y^{s}) = (y_{0}^{u}, y_{0}^{s}) and z = (z^{u}, z^{s}) = (z_{0}^{u}, z_{0}^{s}). We also denote f_{t}^{−i}y = y_{i} = (y_{i}^{u}, y_{i}^{s}) and f_{t}^{−i}z = z_{i} = (z_{i}^{u}, z_{i}^{s}).

Lemma 4.1. Let δ > 0 be sufficiently small. For any x ∈ S^{+} \ W_{δ^{+}}^{u}(0, f_{t}), if

d(x, W_{δ^{+}}^{u}(0, f_{t})) → 0,

then

d(f_{t}^{−n}(x), W_{δ}^{s}(0, f_{t})) → 0,

and the convergence is uniform for all x ∈ S^{+} and t ∈ I, where n = n(x) is the largest integer such that f_{t}^{−1}x, ⋯, f_{t}^{−n}x ∈ S.

Proof. First, we note that for any z ∈ S^{+} ∩ W_{δ^{+}}^{u}(0, f_{t}), f_{t}^{−j}z → 0 uniformly with z and t as j → ∞, since this is true for t = 0 and f_{t}^{−j}z converges to 0 faster than f_{0}^{−j}z does.

To consider the case x ∈ S^{+} \ W_{δ^{+}}^{u}(0, f_{t}), we claim that for any ŷ and ẑ, |ŷ^{s}| > |ẑ^{s}| and |ŷ^{u}| ≤ |ẑ^{u}| imply |ŷ_{1}^{u}| ≤ |ẑ_{1}^{u}|. Otherwise, there would be a point x̂ in a curve joining ŷ and ẑ near which |(f_{t}^{−1}x̂)^{u}| increases with |x̂^{s}|. It contradicts the fact that |(f_{t}^{−1}x)^{u}| decreases with |x^{s}| for x = (x^{u}, x^{s}).

This claim implies inductively that for y = x = (x^{u}, x^{s}) and z = (x^{u}, 0), we always have |y_{i}^{u}| ≤ |z_{i}^{u}| for all 0 ≤ i ≤ n. So if x ∈ S^{+} \ W_{δ^{+}}^{u}(0, f_{t}), we have d(x_{n}, W_{δ}^{s}(0, f_{t})) ≤ d(z_{n}, W_{δ}^{s}(0, f_{t})). Note that f_{t}^{−1} is expanding at a constant rate in the stable direction, we get n = n(x) → ∞ uniformly as d(x, W_{δ^{+}}^{u}(0, f_{t})) → 0. Hence the result of the lemma follows. □

Lemma 4.2. Given any δ > 0, there exist J_{0} > 0 independent of t such that for any x ∈ S^{+}, y, z ∈ W^{u}(x, f_{t}) ∩ S^{+},

(4.1) log \frac{|det Df_{t}^{−n}(z)|_{E_{z}^{u}(f_{t})}|}{|det Df_{t}^{−n}(y)|_{E_{y}^{u}(f_{t})}|} ≤ J_{0}d(y, z),

whenever f_{t}^{−1}x, ⋯, f_{t}^{−n}x ∈ S.

Proof. We assume that δ is small enough such that S^{+} ⊂ Ω_{0}. So we can use (2.5) for the map f_{t}.

We only need to consider the case when t is small since f_{t} is uniformly hyperbolic for all t away from 0.

Further, we only need to consider the case when x is sufficiently close to W_{δ^{+}}^{u}(0, f_{t}), since otherwise, the time that the backward orbit of x leaves the set S is bounded, and the inequality (4.1) surely holds. Further, we may assume that n is the largest integer such that f_{t}^{−1}x, ⋯, f_{t}^{−n}x ∈ S.

Since the angles between E_{y_{i}}^{u}(f_{t}), E_{z_{i}}^{u}(f_{t}) and ℝ^{n^{u}} are exponentially decreasing as i change from n to 0, we just need to show

(4.2) log \frac{|det Df_{t}^{−n}(z)|_{ℝ^{n^{u}}(f_{t})}|}{|det Df_{t}^{−n}(y)|_{ℝ^{n^{u}}(f_{t})}|} ≤ J_{0}d(y, z).

By (2.5), we have

Df_{t}(x)|_{ℝ^{n^{u}}} = (ϕ_{t}(|x^{u}|) + |x^{s}|^{2})I + x^{u} · ϕ′_{t}(|x^{u}|)|x^{u}|^{−1}(x^{u})^{T} ,

where I is the n^{u}×n^{u} identity matrix, and (x^{u})^{T} is the transpose of the column vector x^{u}. Using the fact that det(aI + bxx^{T}) = a^{n−1}(a + b|x|^{2}) for an n^{u} × n^{u} matrix aI + bxx^{T}, we have

(4.3) | det Df_{t}(x)|_{ℝ^{n^{u}}}| = (ϕ_{t}(|x^{u}|)+|x^{s}|^{2})^{n^{u}−1}(ϕ_{t}(|x^{u}|)+|x^{s}|^{2}+ϕ′_{t}(|x^{u}|)|x^{u}|).

It means that | det Df_{t}(x)|_{ℝ^{n^{u}}}| only depends on the norms |x^{u}| and |x^{s}|.

Case One: y^{u}, z^{u} and p^{u} = 0^{u} are on the same line: z^{u} = ay^{u} for some 0 < a < 1.

Note that |z_{n}^{s}| is close to δ by the choice of n. So |z_{n}^{s}| − |z_{n−1}^{s}| is close to (1−κ^{s})δ. By the previous lemma we know that if d(x, W_{δ^{+}}^{u}(0, f_{t})) is sufficiently small, then all x_{n}, y_{n}, z_{n} are close to W_{δ^{+}}^{s}(0, f_{t}) such that

||y_{n}^{s}| − |z_{n}^{s}|| ≤ d(y_{n}, z_{n}) < |z_{n}^{s}| − |z_{n−1}^{s}|,

and the amount d(x, W_{δ^{+}}^{u}(0, f_{t})) can be taken to be independent of x and t. Then we get |y_{n}^{s}| > |z_{n−1}^{s}|. By (2.5), it implies |y_{i}^{s}| > |z_{i−1}^{s}| for all 1 ≤ i ≤ n and in particular, |y_{1}^{s}| > |z^{s}|.

Since y_{1} ∈ S, we have |y_{1}^{u}| ≤ δ by the definition of S. Also, since z ∈ S^{+}, we have |z^{u}| > δ ≥ |y_{1}^{u}|. Therefore, by the fact |y_{i}^{s}| > |z_{i−1}^{s}| and |y_{1}^{u}| ≤ |z^{u}|, we can use the claim of the proof in the previous lemma with ŷ = y_{i} and ẑ = z_{i−1} to get inductively that |y_{i}^{u}| ≤ |z_{i−1}^{u}| for all 1 ≤ i ≤ n. It follows that

∑_{i=0}^{j}(|y_{i}^{u}| − |z_{i}^{u}|)=|y_{0}^{u}| − (|z_{0}^{u}| − |y_{1}^{u}|) − ⋯ − (|z_{j−1}^{u}| − |y_{j}^{u}|) − |z_{j}^{u}| < |y_{0}^{u}| ≤ δ^{+}

for any 0 ≤ j ≤ n.

Now we refine the estimates. Let

τ = {x̂ = (x̂^{u}, x̂^{s}) ∈ W_{δ}^{u}(x, f_{t}) : x̂^{u} = by^{u}, a ≤ b ≤ 1},
τ̄ = {x̄ = (x̄^{u}, x̄^{s}) ∈ W_{δ}^{u}(x, f_{t}) : x̄^{u} = by^{u}, b > 0, δ ≤ |x̄^{u}| ≤ δ^{+}},

and τ_{j} = f_{t}^{−j}τ. Both τ and τ̄ are curves in the unstable manifold W_{δ^{+}}^{u}(x, f_{t}) whose u-component are on the line segment from z^{u} to y^{u} and from δy^{u}/|y^{u}| to δ^{+}y^{u}/|y^{u}| respectively. We denote by ȳ and z̄ the endpoints of the curve τ̄.

Let π^{u} : Ω_{1} → ℝ^{n^{u}} be the projection, and denote τ^{u} = π^{u}τ and τ_{j}^{u} = π^{u}τ_{j}. We know that the first component of the map f_{t}^{j} sends τ_{j}^{u} to τ^{u}. Also recall that z^{u} and y^{u} are in the same direction. By (2.5) we know that all z_{j}^{u} and y_{j}^{u} are in the same direction as well. Hence τ_{j}^{u} is a segment of a straight line. Let E ⊂ ℝ^{n^{u}} be the one dimensional subspace containing (y^{u}, 0^{s}) and (z^{u}, 0^{s}), and let ℓ denote the arc length. We get

||y_{j}^{u}| − |z_{j}^{u}|| = ℓ(τ_{j}^{u}) = ∫_{τ^{u}} ∥Df_{t}^{−j}|_{E}∥dℓ. (4.4)

Also note that Df_{t} preserves the subspace E ⊕ ℝ^{n^{s}}. Hence Df_{t}^{j}(x)|_{E} = Df_{t}(f_{t}^{j−1}x)|_{E} ⋯ Df_{t}(x)|_{E}. By the fact that ∥Df_{t}^{−1}|_{E}∥ is uniformly bounded away from 0, and the C^{2} norm of f_{t} is uniformly bounded from above on Ω_{0}, we have that for j ≤ n,

(4.5)  log \frac{∥Df_{t}^{−j}(z)|_{E}∥}{∥Df_{t}^{−j}(y)|_{E}∥} = log ∏_{i=0}^{j−1}(1 + \frac{∥Df_{t}^{−1}(z_{i})|_{E}∥−∥Df_{t}^{−1}(y_{i})|_{E}∥}{∥Df_{t}^{−1}(y_{i})|_{E}∥})

≤ C ∑_{i=0}^{j−1}|∥Df_{t}^{−1}(z_{i})|_{E}∥−∥Df_{t}^{−1}(y_{i})|_{E}∥| ≤ CD ∑_{i=0}^{j−1}||y_{i}^{u}| − |z_{i}^{u}|| ≤ I_{0},

where we take I_{0} = δ^{+}CD. Since this is true for any y, z ∈ τ̄, by (4.4) we obtain

\frac{||y_{j}^{u}| − |z_{j}^{u}||}{ℓ(τ^{u})} ≤ e^{I_{0}} \frac{||ȳ_{j}^{u}| − |z̄_{j}^{u}||}{ℓ(τ̄^{u})}, j = 0, 1, ⋯, n.

Note that ℓ(τ̄^{u}) = δ^{+}−δ only depends on δ, ℓ(τ^{u}) = d(y^{u}, z^{u}), and ∑_{i=0}^{j}(|ȳ_{i}^{u}|− |z̄_{i}^{u}|) ≤ δ^{+}. We get

∑_{i=0}^{j}||y_{j}^{u}| − |z_{j}^{u}|| ≤ \frac{e^{I_{0}}δ^{+}}{δ^{+} − δ} d(y^{u}, z^{u}) ≤ I_{1}d(y^{u}, z^{u})

for some I_{1} = I_{1}(δ) independent of t.

Now repeating the arguments as for (4.5), we get

(4.6) log \frac{|det Df_{t}^{−j}(z)|_{ℝ^{n^{u}}} |}{|det Df_{t}^{−j}(y)|_{ℝ^{n^{u}}} |} ≤ CD ∑_{i=0}^{j−1} d(y_{i}, z_{i}) ≤ J′d(y^{u}, z^{u}),

where J′ = J′(δ) is independent of t.

Case Two: |y^{u}| = |z^{u}|.

We may assume that |y^{s}| ≤ |z^{s}|. By (4.3), we know that | det Df_{t}(x)^{−1}|_{ℝ^{n^{u}}}| is at least quadratic in terms of |x^{s}|. Thus, we have that there is c_{1} > 0 of order |z^{s}| such that if |y^{u}| = |z^{u}|, then

|det Df_{t}(z)^{−1}|_{ℝ^{n^{u}}} − det Df_{y}(y)^{−1}|_{ℝ^{n^{u}}} | ≤ c_{1}(|z^{s}| − |y^{s}|),

and therefore

(4.7) log \frac{|det Df_{t}(z)^{−1}|_{ℝ^{n^{u}}} |}{| det Df_{t}(y)^{−1}|_{ℝ^{n^{u}}} |} ≤ c_{1}C(|z^{s}| − |y^{s}|) ≤ C_{2}(|z^{s}| − |y^{s}|).

Also, by (2.5), there is a constant c_{0} > 0 of order |y^{u}||z^{s}| such that

||z_{1}^{u}| − |y_{1}^{u}|| ≤ c_{0}(|z^{s}| − |y^{s}|).

Take z̃_{1} = (z̃_{1}^{u}, z̃_{1}^{s}) ∈ W_{δ}^{u}(x_{1}, f_{t}) such that z̃_{1}^{u} = (|y_{1}^{u}|/|z_{1}^{u}|)z_{1}^{u}. That is, z̃_{1} is the point on W^{u}(x_{1}, f_{t}) whose u-coordinate z̃_{1}^{u} is proportional to z_{1}^{u} and satisfies |z̃_{1}^{u}| = |y_{1}^{u}|. Then we denote ỹ_{1} = y_{1}. Hence, by (4.6) we have

log \frac{|det Df_{t}^{−n−1}(z_{1})|_{ℝ^{n^{u}}} |}{| det Df_{t}^{−n−1}(z̃_{1})|_{ℝ^{n^{u}}} |} ≤ J′(|z_{1}^{u}| − |y_{1}^{u}|) ≤ C_{3}(|z^{s}| − |y^{s}|),

where C_{3} = c_{0}J′. By (4.7) and the fact ỹ_{1} = y_{1}, this inequality implies

log \frac{|det Df_{t}^{−n}(z)|_{ℝ^{n^{u}}} |}{| det Df_{t}^{−n}(y)|_{ℝ^{n^{u}}} |} ≤ log \frac{|det Df_{t}^{−n−1}(z̃_{1})|_{ℝ^{n^{u}}} |}{| det Df_{t}^{−n−1}(ỹ_{1})|_{ℝ^{n^{u}}} |} + (C_{2} + C_{3})(|z^{s}| − |y^{s}|).

We denote ỹ_{0}^{s} = y^{s}, z̃_{0}^{s} = z^{s} and choose z̃_{i} and ỹ_{i} in a similar way as we choose z̃_{1} for i = 2, 3, ⋯, n − 1, depending on whichever is larger between |z̃_{i−1}^{s}| and |ỹ_{i−1}^{s}|. Inductively, we have

(4.8) log \frac{|det Df_{t}^{−n}(z)|_{ℝ^{n^{u}}} |}{| det Df_{t}^{−n}(y)|_{ℝ^{n^{u}}} |} ≤ (C_{2} + C_{3}) ∑_{i=0}^{n−1}||z̃_{i}^{s}| − |ỹ_{i}^{s}||.

We observe that all f_{t}^{−n+i}z̃_{i} and f_{t}^{−n+i}ỹ_{i} are in W_{δ}^{u}(x_{n−1}, f_{t}) for 0 ≤ i ≤ n − 1. It is clear that

d((f_{t}^{−n+i}ỹ_{i})^{u}, (f_{t}^{−n+i}z̃_{i})^{u}) ≤ d(y, z).

So there is C_{4} > 0 such that

d((f^{−n+i}ỹ_{i})^{s}, (f^{−n+i}z̃_{i})^{s}) ≤ C_{4}d(y, z), 0 ≤ i ≤ n − 1

because d((f^{−n+i}ỹ_{i})^{s}, (f^{−n+i}z̃_{i})^{s}) is dominated by d((f^{−n+i}ỹ_{i})^{u}, (f^{−n+i}z̃_{i})^{u}) on unstable manifolds close to W_{δ}^{u}(p, f_{t}). Since f_{t} is contracting in ℝ^{n^{s}} direction with the rate κ_{t}^{s}, we get that

d(ỹ_{i}^{s}, z̃_{i}^{s}) ≤ C_{4}(κ_{t}^{s})^{n−i}d(y, z).

Hence by (4.8), we get

(4.9) log \frac{|det Df_{t}^{−n}(z)|_{ℝ^{n^{u}}} |}{| det Df_{t}^{−n}(y)|_{ℝ^{n^{u}}} |} ≤ J′′d(y, z).

Case Three: The general case, y, z ∈ W^{u}(x, S^{+}).

We take ỹ = (ỹ^{u}, ỹ^{s}) ∈ W^{u}(x, S^{+}) such that y^{u}, ỹ^{u} and 0^{u} are on the same line and |ỹ^{u}| = |z^{u}|. Then from Case One and Two we get (4.6) and (4.9) with y, z replaced by y, ỹ and ỹ, z respectively. Therefore we obtain (4.2) with J_{0} = J′ + J′′. □

Proposition 4.3. Given any δ > 0, there exist constants δ′ > 0 and J > 1 such that for all t ∈ [0, 1], x ∈ Λ_{f_{t}}, y, z ∈ W^{u}(x, f_{t}) \ S_{δ} with d^{u}(y, z) ≤ δ′,

(4.10)  J^{−1} ≤ \frac{|det Df_{t}^{−n}(z)|_{E_{z}^{u}(f_{t})}|}{|det Df_{t}^{−n}(y)|_{E_{y}^{u}(f_{t})}|} ≤ J  ∀n ≥ 0.

Proof. Using Lemma 4.2 and the fact that f_{t} is uniformly hyperbolic outside P_{δ}(f_{t}), we can get the result by the same arguments identical to that in the proof of Proposition 3.1 in [HY]. □

The inequalities (4.10) immediately lead to the following estimate of the density function of SRB measure on unstable manifolds.

Corollary 4.4. Given any δ > 0, there exist constants δ′ > 0 and J > 1 such that for any t ∈ [0, 1], SRB measure μ = μ_{t} of f_{t}, and partition ξ = ξ_{t} subordinate to unstable manifolds of f_{t}, the density functions ρ_{x,t} of the conditional measures μ_{x}^{ξ} of μ with respect to Lebesgue measure m_{x}^{u} satisfies

J^{−1} ≤ \frac{ρ_{x,t}(z)}{ρ_{x,t}(y)} ≤ J

for μ-a.e. x ∈ Λ_{f_{t}}, and for all y, z ∈ ξ(x) \ S_{δ} with d^{u}(y, z) < δ′.

Proof. It is well known (see e.g. [L]) that the density function ρ_{x,t} satisfies

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

for all y, z ∈ W^{u}(x, f_{t}). Then we use Proposition 4.3. □

We mention here that we can always increase δ′ by increasing J.

5. Proofs of Theorems A and B

Consider a general map f that has a hyperbolic invariant set Λ.

The map f has local product structure, that is, for any ϵ > 0, there exists δ > 0 such that for any y, z ∈ Λ with d(y, z) ≤ δ, [y, z] = W_{ε}^{s}(y) ∩ W_{ε}^{u}(y) contains exact one point.

A rectangle R in Λ is a set such that y, z ∈ R implies [y, z], [z, y] ∈ R. If 𝒟^{u} and 𝒟^{s} are pieces of W^{u} and W^{s} leaves for f respectively, then [𝒟^{u}, 𝒟^{s}] denotes the rectangle {[y, z] : y ∈ 𝒟^{u}, z ∈ 𝒟^{s}} whenever everything makes sense. If R is a rectangle and x ∈ R, we let

W^{u}(x, R) = W^{u}(x, R, f) = W_{δ}^{u}(x, f) ∩ R,
W^{s}(x, R) = W^{s}(x, R, f) = W_{δ}^{s}(x, f) ∩ 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).

A Markov partition of Λ for f is a set of finite number of rectangles {R_{i}} on Λ with \overline{int R_{i}} = R_{i}, such that (a) int R_{i} ∩ int R_{j} = ∅ whenever i ≠ j, and (b) fR_{i} u-crosses R_{j} and f^{−1}R_{j} s-crosses R_{i} whenever fR_{i} ∩ R_{j} ≠ ∅.

Now we assume that f is as in Theorem A that has a hyperbolic attractor Λ_{f}.

Take a Markov partition ℛ = {R_{i}} of Λ_{f}. We assume that the fixed point p near which we deformed the map is contained in the interior int R_{0} of an element R_{0} of the Markov partition, and that Ω_{1} is small enough such that Ω_{1} ⊂ int R_{0}. Hence, the perturbation is inside int R_{0}. If there is no fixed point contained in the interior of any element of the Markov partition, we can choose a periodic orbit such that every point in the orbit is contained in the interior of some R_{i} of the Markov partition, and make the same perturbation near every point in the orbit.

Denote ∂^{s}R_{i} = {x ∈ R_{i} : x ∉ int W^{u}(x, R_{i})}, and ∂^{s}ℛ = ∪_{i}∂^{s}R_{i}. The properties of Markov partition imply that f(∂^{s}ℛ) ⊂ ∂^{s}ℛ. Since ∂^{s}ℛ consists of stable manifolds of f, and is not perturbed when we construct f_{t}, it also consists of stable manifolds of f_{t} for all t ∈ [0, 1]. Similarly, ∂^{u}ℛ consists of unstable manifolds of f_{t} for all t. Hence, ℛ is a Markov partition for all f_{t}.

Take δ > 0 small enough such that B^{u}(p, δ) ⊂ W^{u}(p, R_{0}). Let

P_{t} = [B^{u}(p, δ), W^{s}(p, R_{0})],

the rectangle determined by B^{u}(p, δ) and W^{s}(p, R_{0}). Clearly, P_{t} ⊂ R_{0}, and W^{s}(x, R_{0}, f_{t}) = W^{s}(x, P_{t}, f_{t}) if x ∈ P_{t}, and W^{s}(x, R_{0}, f_{t}) ∩ P_{t} = ∅ otherwise.

Denote

Q_{t} = f_{t}^{−1}P_{t} \ P_{t}.

Since f_{t} has a hyperbolic attractor Λ_{f_{t}} on which f_{t} is topologically transitive, f_{t} has a unique SRB measure μ_{t} on Λ_{f_{t}} for all 0 < t ≤ 1. Let

ν_{t} = \frac{1}{μ_{t}(Λ_{f_{t}} \ P_{t})}μ_{t}. (5.1)

We have ν_{t}(Λ_{f_{t}} \ P_{t}) = 1.

Lemma 5.1. There is c > 0 such that ν_{t}(Q_{t}) > c for all 0 < t ≤ 1.

Proof. Suppose there exists {t_{n}} ∈ (0, 1] such that ν_{t_{n}}(Q_{t_{n}}) → 0.

Since each f_{t} is topologically transitive on Λ_{f_{t}}, for each rectangle R_{i}, i > 0, there is k = k(i) ≥ 0 independent of t such that f_{t}^{k}R_{i} ∩ Q_{t} ≠ ∅. Note that if f_{t}^{k′}R_{i} ∩ R_{0} ≠ ∅ for some k′ > 0, then f_{t}^{k′}R_{i} u-cross R_{0} by the properties of Markov partition, and therefore f_{t}^{k′−j}R_{i} u-crosses Q_{t} for some 1 ≤ j ≤ k′. So we may assume that k(i) is chosen in such a way that f_{t}^{l}R_{i} ∩ R_{0} = ∅ for all l = 1, ⋯, k.

Since the hyperbolicity of f_{t} is uniform outside R_{0} for all 0 < t ≤ 1, we know that there is c_{1} = c_{1}(i) > 0 independent of t such that

m^{u}(W^{u}(x, f_{t}^{−k}Q_{t}, f_{t})) ≥ c_{1} · m^{u}(W^{u}(x, R_{i}, f_{t}))

for all x ∈ f_{t}^{−k}Q_{t} ∩ R_{i}.

Denote by ξ = ξ_{t} the partition of Λ_{f_{t}} into {W^{u}(x, R_{i}, f_{t}) : x ∈ Λ_{f_{t}}, R_{i} ∈ ℛ}. So ξ_{t} is a partition subordinate to unstable manifolds of f_{t}. Denote by ν_{x,t}^{ξ} the corresponding conditional measure of μ_{t} on ξ(x), where ξ(x) is the element of ξ containing x. Note that ν_{t} and μ_{t} have the same conditional measure on ξ(x). Since μ_{t} is an SRB measure, we can denote by ρ_{x,t} the density function of ν_{x,t}^{ξ} with respect to the Lebesgue measure m_{x}^{u} on W^{u}(x, R_{i}, f_{t}). We know that by Corollary 4.4 the ratio ρ_{x,t}(y)/ρ_{x,t}(z) is bounded away from 0 and infinity for any y, z ∈ W^{u}(x, R_{i}, f_{t}), and the bounds can be chosen in a way that is independent of 0 < t ≤ 1 and x. So we know that there is c_{2} > 0 such that

ν_{x,t}^{ξ}(W^{u}(x, f^{−k}Q_{t}, f_{t})) ≥ c_{2} · ν_{x,t}^{ξ}(W^{u}(x, R_{i}, f_{t})).

Consequently, by invariance of ν_{t}, we get

ν_{t}(Q_{t}) ≥ ν_{t}(f_{t}^{−k}Q_{t} ∩ R_{i}) ≥ c_{2} · ν_{t}(R_{i}).

It follows

ν_{t_{n}}(Q_{t_{n}}) ≥ c_{2} · ν_{t_{n}}(R_{i}).

Note that P_{t} ⊂ R_{0}. Suppose that j = j_{t} is the smallest integer such that f_{t}^{j}P_{t} u-crosses R_{0}. Since f_{t} is uniformly hyperbolic outside P_{t}, {j_{t}} has a upper bound, we again denote it by j. Suppose f_{t}R_{0} is contained in R_{i_{1}} ∪ ⋯ ∪ R_{i_{s}} for some s. Then it is easy to see by invariance of measure ν_{t} that

ν_{t}(R_{0} \ P_{t}) ≤ j ∑_{l=1}^{s} ν_{t}(R_{i_{l}}).

Since

ν_{t}(∪_{i≠0}R_{i}) + ν_{t}(R_{0} \ P_{t}) = 1

for all 0 < t ≤ 1, we know that ν_{t}(∪_{i≠0}R_{i}) is uniformly bounded away from 0 for all t. Therefore, there is 0 < c_{3} < 1 and i such that ν_{t_{n}}R_{i} ≥ c_{3} for infinitely many n. By taking a subsequence we may think that this is true for every n. Hence we get

ν_{t_{n}}(Q_{t_{n}}) ≥ c_{2}c_{3} > 0.

It is a contradiction. □

Put

P_{0,t} = P_{t}, P_{i,t} = f_{t}^{−1}P_{i−1,t} ∩ P_{i−1,t} and
Q_{i,t} = {x ∈ Q_{t} : f_{t}^{j}x ∈ P_{t} for j = 1, ⋯, i} ∀i = 1, 2, · · · .

Lemma 5.2. There is C > 0 such that for all t < (k_{0} + i)^{−β},

\frac{1}{C(k_{0} + i)^{βn^{u}}} ≤ ν_{t}(Q_{i,t}) ≤ \frac{C}{(k_{0} + i)^{βn^{u}}} .

where k_{0} is a real number satisfies δ = k_{0}^{−β}.

Proof. By (2.4) and the definition of f_{t}, we know that for t < (k_{0} + i)^{−β}, W^{u}(p, P_{i,t}) is a ball of radius (i+k_{0})^{−β} on W_{δ}^{u}(p). So m^{u}(W^{u}(p, P_{i,t})) = C_{1}(i+ k_{0})^{−βn^{u}}, where C_{1} is equal to the volume of the unit ball in n^{u} dimensional Euclidean space.

For any x ∈ Q_{i,t}, consider the holonomy that θ : W^{u}(x, Q_{t}, f_{t}) → W^{u}(x, P_{t}) maps W^{u}(x, Q_{i,t}, f_{t}) to W^{u}(p, P_{i,t}). By Proposition 3.4, there exists J_{1}^{s} ≥ 1 such that

\frac{1}{J_{1}^{s}} ≤ \frac{m^{u}(W^{u}(x, Q_{i,t}, f_{t}))}{m^{u}(W^{u}(p, P_{i,t}))} ≤ J_{1}^{s}.

Now by Corollary 4.4, the ratio ρ_{x,t}(y)/ρ_{x,t}(z) of the density of the conditional measure of ν_{x,t}^{ξ} at any two points y, z ∈ W^{u}(x, Q_{i,t}, f_{t}) are uniformly bounded by a constant J > 0. So we get the result. □

Proof of Theorem A.

By (1.1), we only need to show that

∫ log | det Df_{t}|_{E^{u}(f_{t})}|dμ_{t} → 0 as t → 0.

Take ε > 0. Since Df_{0}(p)|_{E_{p}^{u}(f_{0})} = id, there is a constant t′ > 0 and a neighborhood V of p such that log | det Df_{t}(x)|_{E_{x}^{u}(f_{t})}| ≤ ε/2 for all x ∈ V and t ∈ (0, t′].

It is easy to see that there is δ^{∗} > 0 such that V contains the rectangle [W_{δ^{∗}}^{s}(p), W_{δ^{∗}}^{u}(p)] for all 0 < t ≤ 1. Take j > 0 such that f_{t}^{j}W^{s}(p, P_{t}) ⊂ W_{δ^{∗}}^{s}(p) for all t ∈ (0, 1]. This is possible since f_{t} is uniformly contracting along stable direction. Also, take some k > 0 such that (k_{0}+k)^{−β} ≤ δ^{∗}. Since W^{u}(p, P_{k,t}) ⊂ W_{δ^{∗}}^{u}(p) for all t, We have f_{t}^{j}P_{k+j,t} ⊂ V.

Note that Q_{1,t} is the set of points that enter P_{t} under the map f_{t}, and P_{t}\P_{1,t} is the set of points x that leave P_{t} under f_{t}. We have Q_{1,t} and P_{1,t} are disjoint and

Q_{1,t} ∪ P_{1,t} = f_{t}^{−1}(P_{t}); P_{t} = (P_{t} \ P_{1,t}) ∪ P_{1,t}.

Since ν_{t}(f_{t}^{−1}P_{t}) = ν_{t}(P_{t}), we have ν_{t}(Q_{1,t}) = ν_{t}(P_{t} \ P_{1,t}). Similarly, we have ν_{t}(Q_{i,t}) = ν_{t}(P_{i−1,t} \ P_{i,t}). Hence,

ν_{t}(Λ_{f_{t}} \ P_{k+j,t}) = ν_{t}(Λ_{f_{t}} \ P_{t}) + ∑_{i=1}^{k+j−1} ν_{t}(Q_{i,t}).

By the definition of ν_{t} we know that ν_{t}(Λ_{f_{t}} \ P_{t}) = 1 for all 0 < t ≤ 1 and by Lemma 5.2, ∑_{i=1}^{k+j−1} ν_{t}(Q_{i,t}) is uniformly bounded for all 0 < t ≤ 1 since the selections of k, j are independent of t.

Also note that βn^{u} < 1. So the series ∑_{i=1}^{∞} (i + k_{0})^{−βn^{u}} diverges. Therefore we can choose N > 0 large enough such that if all ν_{t}(Q_{i,t}), 1 ≤ i ≤ N, satisfy the estimates in Lemma 5.2, then

\frac{ε}{2D} ∑_{i=k+j}^{N} ν_{t}(Q_{i,t}) ≥ ν(Λ_{f_{t}} \ P_{k+j,t}),

where D = max{log | det Df_{t}(x)|_{E_{x}^{u}(f_{t})}| : x ∈ Λ_{f_{t}}, t ∈ (0, 1]}. Then we take

0 < t′′ ≤ (N + k_{0})^{−β}.

So for any t ∈ (0, t′′], the above inequality holds, and therefore,

\frac{ε}{2D}ν_{t}(P_{k+j,t}) = \frac{ε}{2D} ∑_{i=k+j}^{∞} ν_{t}(Q_{i,t}) ≥ ν_{t}(Λ_{f_{t}} \ P_{k+j,t}).

By (5.1), μ_{t} = [μ_{t}(Λ_{f_{t}} \ P_{t})]^{−1}ν_{t}, the above inequality also holds for μ_{t} if t ∈ (0, t′′]. Since

μ_{t}(P_{k+j,t}) + μ_{t}(Λ_{f_{t}} \ P_{k+j,t}) = 1,

we get μ_{t}(Λ_{f_{t}} \ P_{k+j,t}) ≤ \frac{ε}{2D}. It follows

μ_{t}(Λ_{f_{t}} \ f_{t}^{j}P_{k+j,t}) ≤ \frac{ε}{2D}. (5.2)

Recall f_{t}^{j}P_{k+j,t} ⊂ V, in which log | det Df_{t}(x)|_{E_{x}^{u}(f_{t})}| ≤ ε/2 for t ∈ (0, t′). We get that if 0 < t ≤ min{t′, t′′}, then

∫ log | det Df_{t}|_{E^{u}(f_{t})}|dμ_{t}
= ∫_{Λ_{t}\f_{t}^{j}P_{k+j,t}} log | det Df_{t}|_{E^{u}(f_{t})}|dμ_{t} + ∫_{f_{t}^{j}P_{k+j,t}} log | det Df_{t}|_{E^{u}(f_{t})}|dμ_{t}
≤ D · μ_{t}(Λ_{t} \ f_{t}^{j}P_{k+j,t}) + \frac{ε}{2} · μ_{t}(f_{t}^{j}P_{k+j,t}) ≤ D · \frac{ε}{2D} + \frac{ε}{2} = ε.

We proved the theorem. □

Proof of Corollary.

By the proof of Theorem A, we know that for any neighborhood V of p, there are constants k, j > 0 such that f_{t}^{j}P_{k+j,t} ⊂ V. Also, for any ε > 0 we can find t′′ > 0 such that for all t ∈ (0, t′′], (5.2) holds. It means that μ_{t}(V) ≥ μ_{t}(f_{t}^{j}P_{k+j,t}) → 1 as t → 0. This implies that μ_{t} → δ_{p}. The rest result of the corollary is directly from Theorem A. □

Proof of Theorem B.

The proof of the homotopy follows the construction of f_{t}. The proof that f_{0} has an infinite SRB measure is the same as in [HY] and thus omitted. □

Remark. A referee pointed out that one can obtain Theorem A easily if nonexistence of SRB measures for f_{0} is assumed. In fact, after the construction of the map f_{t}, one can take a weak limit of a subsequence μ_{t_{i}} of the SRB measures for f_{t_{i}} as t_{i} → 0, and get an f_{0} invariant measure μ′. Using the fact that there is a common generator ξ for all f_{t} such that h_{μ_{t}}(f_{t}) = h_{μ_{t}}(f_{t}, ξ), one can show that lim sup_{n→0} h_{μ_{t_{i}}}(f_{t_{i}}) ≤ h_{μ′}(f_{0}). It implies that h_{μ′}(f_{0}) ≥ ∫ log | det Df_{0}|_{E^{u}}|dμ′. Hence, μ′ satisfies entropy formula by Ruelle inequality, and therefore, if μ′ has positive Lyapunov exponents, then it must be an SRB measure following a result of Ledrappier and Young ([LY]), which contradicts the assumption. So μ′ does not have positive Lyapunov exponents, and hence h_{μ′}(f_{0}) = 0.

As stated in Remark 1.4, Theorem B implies that f_{0} does not admits SRB measures. Thus, Theorem A can be proved by the arguments above by using Theorem B. In particular, for Anosov systems on the two dimensional torus, nonexistence of SRB measures follows from a result of Hu and Young ([LY]).

Acknowledgment. The authors thank Dmitry Dolgopyat, Marcelo Viana, Mark Pollicott, and David Ruelle for many fruitful discussions on this and other related problems. The authors also thank referees for constructive comments and suggestions.

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Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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

Department of Mathematics, Wake Forest University, Winston Salem, NC 27106, USA

  E-mail address: jiangm@wfu.edu

Department of Mathematics, the Graduate Center and Queens College of CUNY, Flushing, NY 11367, USA

  E-mail address: yunping.jiang@qc.cuny.edu