DIMENSIONS OF INVARIANT SETS OF EXPANDING MAPS

Huyi Hu *

Department of Mathematics, University of Maryland, College Park, MD 20742

Abstract. We consider a compact invariant set Λ of an expanding map of a manifold M and give upper and lower bounds for the Hausdorff Dimension dim_{H}(Λ), and box dimensions \underline{dim}_{B}(Λ) and \overline{dim}_{B}(Λ). These bounds are given in terms of the topological entropy, topological pressure, and uniform Lyapunov exponents of the map.

A measure-theoretic version of these results is also included.

Key words. Hausdorff dimension, box dimension, expanding, entropy, Lyapunov exponent

AMS(MOS) subject classifications. 58F15, 58F11, 28D20

§0 Introduction

It is well known that many self-similar sets, or fractals, can be realized as invariant sets of smooth expanding maps. The purpose of this paper is to give estimates on the box dimensions and Hausdorff dimensions of these sets.

Most of the known results on dimensions in dynamical systems are of the following types. Some are about the dimensions of invariant measures (see e.g. [L] and [LY]). Some are about the dimensions of sets constructed via a sequence of affine contractions (e.g. [F1] and [GL]). Others deal with subsets of the line. See also [G], [MM], [BU], etc.

At present, there are few known effective ways to calculate the box and Hausdorff dimensions of sets (sets, not measures) invariant under nonlinear maps in dimensions larger than 1. In this paper we consider invariant sets of uniformly expanding maps. We give upper and lower bounds for box and Hausdorff dimensions in terms of topological entropy, topological pressure, and uniform Lyapunov exponents. Examples of self-affine sets show that these bounds are sharp (see §1, Example 2 ). We will also give parallel results for dimensions of invariant measures.

We state our results.

Let M be an m-dimensional C^{∞} Riemannian manifold with Riemannian measure ν, and f be a C^{2} map from M to itself. Let Λ ⊂ M be an f-invariant compact subset, where f-invariance means fΛ = Λ. We also assume that f is expanding on Λ, i.e., there exist C > 0, κ > 1, such that ∀ x ∈ Λ, n ≥ 1,

∥Df^{n}(x)u∥ ≥ Cκ^{n}∥u∥ ∀ u ∈ T_{x}M.

Without loss generality we can take C = 1.

* Part of this work was done when I was in the Department of Mathematics, University of Arizona.

We denote by |Df(x)| the Jacobian of Df(x) : T_{x}M → T_{fx}M, by ∥Df(x)∥ the norm of Df(x) and by [|Df(x)|] the minimum norm of Df(x), i.e., [|Df(x)|] = inf_{0≠u∈T_{x}M} \frac{∥Df(x)u∥}{∥u∥}.

Put

σ_{+} = lim_{n→∞} \frac{1}{n} log ∥Df^{n}|_{Λ}∥  and  σ_{−} = lim_{n→∞} \frac{1}{n} log[|Df^{n}|_{Λ}|].

where ∥Df^{n}|_{Λ}∥ = sup_{x∈Λ} ∥Df^{n}(x)∥ and [|Df^{n}|_{Λ}|] = inf_{x∈Λ}[|Df^{n}(x)|]. We call σ_{+} and σ_{−} the

largest and smallest uniform Lyapunov exponents respectively. Since f is expanding on Λ, σ_{−} > 0.

We also denote h_{top} = h(f|_{Λ}), the topological entropy, and P = P (f|_{Λ}, − log |Df(x)|), the topological pressure of f|_{Λ} for the function − log |Df(x)|. The Hausdorff dimension, lower and upper box dimensions of Λ are denoted by dim_{H}(Λ), \underline{dim}_{B}(Λ) and \overline{dim}_{B}(Λ) respectively. Their precise definitions are given in §2.

Theorem A. Let f be a C^{2} self-map of an m-dimensional Riemannian manifold M, and let Λ be a compact f-invariant set. Assume that f is expanding on Λ. Then

i) \frac{h_{top}}{σ_{+}} ≤ D(Λ) ≤ \frac{h_{top}}{σ_{−}}, and

ii) m + \frac{P}{σ_{−}} ≤ D(Λ) ≤ m + \frac{P}{σ_{+}} ,

where D(Λ) is dim_{H}(Λ), \underline{dim}_{B}(Λ), or \overline{dim}_{B}(Λ).

To prove the lower bound for D(Λ), we use part of the following theorem, which is a measure-theoretic version of Theorem A.

For an ergodic measure μ on Λ, we let h = h_{μ}(f) denote the measure-theoretic entropy, λ_{1} and λ_{s} be the largest and smallest Lyapunov exponents, and λ be the sum of all Lyapunov exponents including multiplicities — all of them depend on μ. The definitions related to dimensions of measures are stated in §2.

Theorem B. Let f be a C^{2} self-map of an m-dimensional Riemannian manifold M, and let Λ be a compact f-invariant set. Assume that f is expanding on Λ and μ is an ergodic probability measure on Λ. Then

i) \frac{h}{λ_{1}} ≤ D(μ) ≤ \frac{h}{λ_{s}}, and

ii) m − \frac{λ − h}{λ_{s}} ≤ D(μ) ≤ m − \frac{λ − h}{λ_{1}},

where D(μ) is dim_{H}(μ), \underline{dim}_{B}(μ), \overline{dim}_{B}(μ), \underline{dim}_{L}(μ), or \overline{dim}_{L}(μ).

Remark. Theorem B is a generalization of a known result that if f is conformal then the entropy h_{μ}(f) is equal to the product of the Lyapunov exponent and the Hausdorff dimension (see [R]). Also, it has been conjectured that the formula of Ledrappier and Young for dim_{H}(μ) is true for invariant measures of smooth expanding maps. The results in Theorem B are consistent with this conjecture.

The major work in this paper is to prove part ii) of Theorem B. Proposition 2 in §4 plays a key role for the proof.

§1 Examples

We begin by looking at some examples.

Example 1. Let R_{0} be a unit square in IR^{2}. Take real numbers 1 < a < b. Put n rectangles R_{1}, ⋯, R_{n} of width a^{−1} and height b^{−1} in an arbitrary way, as long as they are separated from each other. Define a C^{2} map f on IR^{2} such that f maps each of R_{i} affinely onto R_{0}. The n inverses of f|_{R_{i}} : R_{i} → R_{0}, i = 1, ⋯, n, determine an f-invariant set, or

fractal, Λ ⊂ ⋃_{i=1}^{n} R_{i} in a usual way (see [H] or [F2]). In fact, Λ = ⋂_{n=0}^{∞} f^{−i}(R_{1} ∪ ⋯ ∪ R_{n}),

which is an invariant set of f.

Clearly, f is expanding on Λ with expanding rates a and b in horizontal and vertical directions, and the smaller and larger uniform Lyapunov exponents of f are σ_{−} = log a and σ_{+} = log b respectively. It is also easy to see that h_{top} = h(f|_{Λ}) = log n. Since − log |Df(x)| = − log a−log b is a constant on Λ, P = P (f|_{Λ}, − log |Df(x)|) = h_{top}(f|_{Λ})− log a − log b = log n − log a − log b (see [W] §9.2). By Theorem A we get that

max\{\frac{log n}{log b}, 1 + \frac{log n − log b}{log a}\} ≤ D(Λ) ≤ min\{\frac{log n}{log a}, 1 + \frac{log n − log a}{log b}\}.  (1.1)

In this case all the terms in the bounds are explicit.

Fig. 1 Fig. 2 In Fig 1, a = 4.1, b = 4.9 and n = 4. Theorem A gives 0.982 ⋯ ≤ D(Λ) ≤ 0.984 ⋯,

which is less than 1. In Fig 2, we use the same a and b, but n = 5. By Theorem A, we get 1.014 ⋯ ≤ D(Λ) ≤ 1.140 ⋯, which is larger than 1. (In these figures the black rectangles

form the set ⋂_{k=0}^{2} f^{−k}( ⋃_{i=1}^{n} R_{i}).)

Example 2 (Generalized Sierpinski carpets). Let R_{0} be a unit square in IR^{2}. Take integers 0 < p < q. Partition R_{0} into p columns and q rows, producing pq rectangles of size p^{−1} × q^{−1}. Select a subcollection of these rectangles R = {R_{1}, ⋯, R_{n}}, and generate an invariant set Λ in the same way as in the last example. (If we identify R_{0} with a torus, then f is a C^{2} map even though some of rectangles are adjacent.) The dimensions of the set Λ were studied by McMullen ([Mc]) and Bedford ([B]). The box and Hausdorff dimensions of Λ are

dim_{B}(Λ) = \frac{log p̂}{log p} + \frac{1}{log q} · log(\frac{1}{p̂} ∑_{j=1}^{p} n_{j}),  dim_{H}(Λ) = \frac{1}{log p} log(∑_{j=1}^{p}(n_{j})^{\frac{log p}{log q}} ), (1.2)

where n_{j} denotes the number of rectangles selected from the jth column and p̂ is the number of columns containing at least one rectangle in R. (See also [F2].) These results are consistent with our estimates.

Note that in these examples the dimensions of Λ depend not only on the number n of selected rectangles but also on the configuration of the rectangles in R. This shows that in general one cannot get an exact formula in terms of σ_{−}, σ_{+}, h_{top} and P for either box dimensions or Hausdorff dimensions.

In fact, by choosing suitable configurations of the rectangles, the bounds in Theorem A can be reached. Let us look at the following cases.

i) Take n_{i} = 0 or 1 for all 1 ≤ i ≤ p. Thus n = p̂. By (1.2) we get that dim_{B}(Λ) =

dim_{H}(Λ) = \frac{log p̂}{log p}, which is one of the upper bounds in (1.1).

ii) For all 1 ≤ i ≤ p, put n_{i} = k. In this case, n = kp. We obtain that dim_{B}(Λ) =

dim_{H}(Λ) = 1 + \frac{log k}{log q}. This coincides the other upper bounds in (1.1).

iii) Choose p̂ = 1, i.e. all selected rectangles are in one column. Hence only one n_{i} is

greater than 0 and n is equal to this n_{i}. (1.2) implies that dim_{B}(Λ) = dim_{H}(Λ) = \frac{log n}{log q},

one of the lower bounds in (1.1).

iv) Let n_{i} be equal to either 0 or q. So n = p̂·q, and dim_{B}(Λ) = dim_{H}(Λ) = 1+ \frac{log p̂}{log p},

same as the other lower bound.

Example 3. (Similitudes) Let Λ be constructed as in Example 2 but put p = q > 0. In this case f is called a similitude (see [H]). Since σ_{+} = σ_{−} = log p, the estimates in Theorem A become equality. Thus, we obtain that

dim_{H}(Λ) = dim_{B}(Λ) = \frac{log n}{log p}.

In general, for any invariant set Λ of a similitude f,

dim_{H}(Λ) = dim_{B}(Λ) = \frac{h_{top}}{log ∥Df|_{Λ}∥}.

§2 Definitions and Notations

First we give the definitions of the various notions of dimensions. (See e.g. [F2].) For a metric space X, its Hausdorff dimension is defined by

dim_{H}(X) = inf{α : lim_{ϵ→0} inf_{diam C<ϵ} ∑_{A∈C}(diam A)^{α} = 0},

where C is a cover of X and diam C = inf{diam A : A ∈ C}. If υ is a probability measure on X, then the Hausdorff dimension of the measure υ is given by

dim_{H}(υ) = inf{dim_{H}(Y) : Y ⊂ X, υY = 1 }.

The upper and lower box dimensions of X are defined by

\overline{dim}_{B}(X) = lim sup_{ϵ→0} \frac{log N(ϵ)}{− log ϵ}  and  \underline{dim}_{B}(X) = lim inf_{ϵ→0} \frac{log N(ϵ)}{− log ϵ}

respectively, where N(ϵ) denotes the minimum number of balls of radius ϵ which cover X. \overline{dim}_{B}(X) and \underline{dim}_{B}(X) are also called upper and lower capacities. If \overline{dim}_{B}(X) = \underline{dim}_{B}(X), then we simply call this number the box dimension, and denoted it by dim_{B}(X). Similarly, we have

\overline{dim}_{B}(υ) = lim_{δ→0} inf\{\overline{dim}_{B}(Y) : Y ⊂ X, υY > 1 − δ\},

and

\underline{dim}_{B}(υ) = lim_{δ→0} inf\{\underline{dim}_{B}(Y) : Y ⊂ X, υY > 1 − δ\}.

F. Ledrappier has introduced the following notions [L]:

\overline{dim}_{L}(υ) = lim_{δ→0} lim sup_{ϵ→0} \frac{log N(ϵ, δ)}{− log ϵ}  and  \underline{dim}_{L}(υ) = lim_{δ→0} lim inf_{ϵ→0} \frac{log N(ϵ, δ)}{− log ϵ},

where N(ϵ, δ) is the minimum number of balls of diameter ϵ covering a subset in X of measure greater than 1 − δ.

The following inequalities are immediate:

dim_{H}(υ) ≤ dim_{H}(X),  (2.1)
\underline{dim}_{L}(υ) ≤ \underline{dim}_{B}(υ) ≤ \underline{dim}_{B}(X), \overline{dim}_{L}(υ) ≤ \overline{dim}_{B}(υ) ≤ \overline{dim}_{B}(X), (2.2)

and

dim_{H}(X) ≤ \underline{dim}_{B}(X) ≤ \overline{dim}_{B}(X).  (2.3)

Moreover, L.-S. Young has proved ([Y]) that

dim_{H}(υ) ≤ \underline{dim}_{L}(υ).  (2.4)

The local entropy of f at x with respect to μ is defined as (see [BK])

h_{μ}(x, f) = lim_{ϵ→0} lim sup_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ)
= lim_{ϵ→0} lim inf_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ).  (2.5)

It satisfies h_{μ}(fx, f) = h_{μ}(x, f) and ∫ h_{μ}(x, f)dμ(x) = h_{μ}(f), where h_{μ}(f) is the measuretheoretic entropy of f. Since μ is ergodic, we can write h = h_{μ}(x, f) μ-a.e. x.

Let

B(x, α) = {y ∈ M : d(y, x) ≤ α}

and

B_{n}(x, α) = {y ∈ M : d(f^{i}y, f^{i}x) ≤ α, i = 0, ⋯, n}.

B(x, α) and B_{n}(x, α) are called the α-ball and (n, α)-ball about x respectively. If Γ ⊂ M, we denote

B(Γ, α) = ⋃_{x∈Γ} B(x, α), B_{n}(Γ, α) = ⋃_{x∈Γ} B_{n}(x, α).

§3 Preliminaries

We denote by f̂: (M̂, μ̂) → (M̂, μ̂) the inverse limit of f : (M, μ) → (M, μ). (For a precise definition, see e.g. [P] under the terminology “natural extension”.) Let π_{i} : M̂ → M be the projection defined by π_{i}({x_{j}}_{j=0}^{∞}) = x_{i} ∀ {x_{j}}_{j=0}^{∞} ∈ M̂. For any x̂ = {x_{i}}, f̂{x_{i}} = {fx_{i}}. The inverse of f̂ is given by f̂^{−1}{x_{i}} = {y_{i}}, where y_{i} = x_{i+1}.

Let

Λ̂ = {x̂ = {x_{i}} ∈ M̂: x_{i} ∈ Λ ∀i ≥ 0}.

Clearly, f̂Λ̂ = Λ̂, and μ̂ is an ergodic measure on Λ̂. It is also easy to see that with respect to cocycle {Df_{x_{0}}^{n}}, for μ̂-a.e. x̂ = {x_{i}} ∈ Λ̂, the Lyapunov exponents are identical to those in the system f : (M, μ) → (M, μ), i.e. λ_{1} > ⋯ > λ_{s}.

Take 0 < γ_{0} ≤ \frac{1}{200m}λ_{s}.

Let ⟨⟨·, ·⟩⟩ be the inner product on T M given by the Riemannian structure which induces the norm ∥ · ∥. Let ⟨·, ·⟩ and | · | denote the usual inner product and norm on IR^{m} respectively. For α > 0, let B̃(α) be the ball in IR^{m} centered at the origin of radius α.

We now introduce the properties of Lyapunov charts {Φ_{x} : x̂ ∈ Λ̂} for f̂. The proof is similar with that in [LY].

Fact 1. ∀γ ≥ 0, there exists a measurable function l : Λ̂ → [1, ∞] with l(f̂^{±1}x̂) ≤ l(x̂)e^{γ} and l(x̂) < ∞ for μ-a.e. x̂ ∈ Λ̂, and a set of embedings Φ_{x} : B̃(l(x̂)^{−1}) → M at each point x̂ ∈ Λ̂ such that the following holds.

i) Φ_{x}(0) = x_{0}, and the preimages R_{i}(x̂) = DΦ_{x}(0)^{−1}(E_{i}(x̂)) of E_{i}(x̂) are mutually orthogonal in IR^{m}, where i = 1, ⋯, s.

ii) Let F_{x} = Φ−1 ; _{f} ; _{x} ∘f ∘Φ_{x} be the connecting map between the chart at x̂ and the chart

at f̂x̂. Then ∀1 ≤ i ≤ s, u ∈ R_{i}(x̂), |u|e^{λ_{i}−γ} ≤ |DF_{x}(0) u| ≤ |u|e^{λ_{i}+γ}.

iii) If L(g) denotes the Lipschitz constant of the function g, then L(F ± ; _{x}−DF ± ; _{x}(0)) ≤ γ,

and L(DF ± ; _{x}) ≤ l(x̂).

iv) For all v, v′ ∈ B̃(l(x̂)^{−1}), K^{−1}d(Φ_{x}v, Φ_{x}v′) ≤ |v − v′| ≤ l(x̂)d(Φ_{x}v, Φ_{x}v′) for some

universal constant K ≥ 1.

We define l : Λ → [1, ∞] by l(x) = inf{l(x̂) : x̂ ∈ π_{0}^{−1}(x)}. There is no ambiguity between l(x) and l(x̂), because the domains are different. Clearly, l(fx) ≤ l(x)e^{γ}.

For any l ≥ 1 let

Λ_{l} = {x ∈ Λ : l(x) < l}.

Proposition 1. If x ∈ Λ_{l} for some l ≥ 1, then ∀n > 0,

B(x, ϵe^{−n(λ_{1}+2γ)}) ⊂ B_{n}(x, ϵKl )  and  B_{n}(x, ϵ) ⊂ B(x, ϵKle^{−n(λ_{s}−3γ)}),

provided 0 < ϵ ≤ \frac{1}{2}l^{−2}e^{−2nγ}. Proof. Take z ∈ B(x, ϵe^{−n(λ_{1}+2γ)}).

Since l(x) < l, we can choose x̂ ∈ π_{0}^{−1}(x) such that l ≥ l(x̂). Denote v = Φ−1 ; _{x}z, then by Fact 1. iv),

|v| ≤ l(x̂)d(x, z) ≤ l(x̂) · ϵe^{−n(λ_{1}+2γ)} ≤ ϵle^{−n(λ_{1}+2γ)}.

Write F (n) ; _{x} = F_{f}^{n}_{x} ∘ ⋯ ∘ F_{f} ; _{x} ∘ F_{x}. By Fact 1. ii) and iii) we have

|F (n) ; _{x}v| ≤ e^{n(λ_{1}+2γ)}|v| ≤ e^{n(λ_{1}+2γ)} · ϵle^{−n(λ_{1}+2γ)} = ϵl ≤ \frac{1}{2}l(x̂)^{−1}e^{−2nγ} ≤ l(f̂^{n}x̂)^{−1}.

So F (n) ; _{x}v ∈ B̃(l(f̂^{n}x̂)^{−1}). Again, by Fact 1. iv),

d(f^{n}x, f^{n}z) ≤ K|F (n) ; _{x}v| ≤ K · ϵl = ϵKl.

It means z ∈ B_{n}(x, ϵKl ).

The second inclusion can be proved similarly.

Recall that Λ is a compact subset in an m-dimensional Riemannian manifold M and ν is the Riemannian measure. Lemma 1. ∃ K_{1} ≥ 1, ϵ_{1} > 0, such that ∀ 0 < ϵ < ϵ_{1}, x ∈ Λ,

K_{1}^{−1}ϵ^{m} ≤ νB(x, ϵ) ≤ K_{1}ϵ^{m}.

Proof. This is clear.

Lemma 2. ∃ K_{2} ≥ 1, ϵ_{2} > 0, such that ∀ 0 < ϵ < ϵ_{2}, x ∈ Λ, n ≥ 0,

K_{2}^{−1}ϵ^{m} ≤ νB_{n}(x, ϵ)|Df^{n}(x)| ≤ K_{2}ϵ^{m}.

Proof. Use distortion estimates J^{−1} ≤ \frac{|Df^{n}(x)|}{|Df^{n}(y)|} ≤ J ∀y ∈ B_{n}(x, ϵ) and Lemma 1.

Lemma 3. ∀ a > 0,

lim sup_{n→∞} −\frac{1}{na} log μB(x, e^{−na})  and  lim inf_{n→∞} −\frac{1}{na} log μB(x, e^{−na}),

as functions of x, are constant for μ-almost every x in Λ. Proof. There exists k > 0 such that ∀ x ∈ Λ, fB(x, e^{−an}) ⊂ B(fx, e^{−a(n−k)}) for all large n. So μB(x, e^{−an}) ≤ μB(fx, e^{−a(n−k)}). This implies both limits are f-invariant and therefore are constant almost everywhere.

Lemma 4. ∃ ϵ_{3} > 0, such that ∀ 0 < ϵ < ϵ_{3} μ-a.e. x ∈ Λ,

h = lim_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ).  (3.1)

Proof. Note that f is expanding on Λ. If ϵ_{3} is sufficiently small, then ∃k > 0 such that for any 0 < ϵ′ ≤ ϵ ≤ ϵ_{3}, B_{k}(x, ϵ) ⊂ B(x, ϵ′) ⊂ B(x, ϵ) ∀x ∈ Λ. Therefore, for any n > 0, B_{n+k}(x, ϵ) ⊂ B_{n}(x, ϵ′) ⊂ B_{n}(x, ϵ) ∀x ∈ Λ. Hence, we have

lim sup_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ) = lim sup_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ′) ∀x ∈ Λ.

This means that the above limit is independent of ϵ provided 0 < ϵ < ϵ_{3}. So, by (2.5) h = lim sup_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ) μ-a.e. x ∈ Λ.

Similarly, we also have that h = lim inf_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ) μ-a.e. x ∈ Λ. The result

then follows.

§4 Proof of Theorem B

Proposition 2 is essential for the proof of the second part of Theorem B. It says that, up to factors e^{−nγ}, the measure of an ϵe^{−nλ_{s}}-ball is greater than e^{−n(mλ_{s}−λ+h)}, and the measure of an ϵe^{−nλ_{1}}-ball is less than e^{−n(mλ_{1}−λ+h)}. The idea of the proof comes from the following observation.

The volumes (i.e. the ν-measure) of each ϵe^{−nλ_{s}}-ball and each (n, ϵ)-ball are roughly e^{−nmλ_{s}} and e^{−nλ} respectively. Therefore an ϵe^{−nλ_{s}}-ball can be covered by ϵe^{−n(mλ_{s}−λ)} (n, ϵ)-balls. Since each “nice” (n, ϵ)-ball gives a contribution of measure at most e^{−nh}, a “nice” ϵe^{−nλ_{s}}-ball should not contain measure more than e^{−n(mλ_{s}−λ+h)}.

Similarly, by comparing the volumes of an ϵe^{−nλ_{1}}-ball and an (n, ϵ)-ball, we get that an (n, ϵ)-ball can be covered by e^{n(mλ_{1}−λ)} balls of radius ϵe^{−nλ_{1}}. Therefore, a general ϵe^{−nλ_{1}}-ball should contribute measure at least e^{−n(mλ_{1}−λ+h)}.

The proof for the proposition carries out the above idea. However, since convergences in (2.5) and (3.1) are not uniform, we have to choose some “nice” sets on which the above argument can be applied. Also, we consider all ϵe^{−nλ_{s}}-balls and (n, ϵ)-balls which cover the “nice” sets simultaneously .

Proposition 2. Let 0 < γ < \frac{1}{10} min\{γ_{0}, h\} and 0 < ϵ < \frac{1}{10} min\{ϵ_{1}, ϵ_{2}, ϵ_{3}, 1\}. Then μ-a.e. x ∈ Λ,

m − \frac{λ − h + (2m + 3)γ}{λ_{s} − 2γ} ≤ lim inf_{n→∞} −\frac{1}{n(λ_{s} − 2γ)} log μB(x, ϵe^{−n(λ_{s}−2γ)}),

lim sup_{n→∞} −\frac{1}{n(λ_{1} + 2γ)} log μB(x, ϵe^{−n(λ_{1}+2γ)}) ≤ m − \frac{λ − h − (2m + 3)γ}{λ_{1} + 2γ} .

Proof of Proposition 2 (The First Inequality).

Write λ_{−} = λ_{s} − 2γ.

Suppose this inequality is not true. Then by Lemma 3, for some small γ and ϵ, μ-a.e. x ∈ Λ,

lim inf_{n→∞} −\frac{1}{nλ_{−}} log μB(x, ϵe^{−nλ_{−}}) < m − \frac{λ − h + (2m + 3)γ}{λ_{−}} .  (4.1)

Fix l > 1 such that Kl^{−1} ≤ 1 and μΛ_{l} > 0.

Put

A′_{n} = {x ∈ Λ : μB_{k}(x, ϵl^{−2}e^{−2kγ} ) < exp[−k(h − γ)] ∀k ≥ n}.

By Lemma 4 lim_{n→∞} μA′_{n} = 1, because μB_{k}(x, ϵl^{−2}e^{−2kγ}) ≤ μB_{k}(x, ϵ) ∀k ≥ 0.

Put

A′′_{n} = {x ∈ Λ : |Df_{x}^{k}| < exp k(λ + γ) ∀k ≥ n}.

By the fact that λ = lim_{n→∞} \frac{1}{n} log |Df^{n}(x)| μ-a.e. x ∈ Λ, lim_{n→∞} μA′′_{n} = 1.

Put

A′′′_{n} = {x ∈ Λ : μB(x, ϵe^{−nλ_{−}}) ≥ exp[−n(mλ_{−} − λ + h − (2m + 3)γ)] }.

By (4.1) μ( ⋃_{n=N}^{∞} A′′′_{n}) = 1 ∀N > 0.

Put A_{n} = A′_{n} ∩ A′′_{n} ∩ A′′′_{n} ∩ Λ_{l}. Since both {A′_{n}} and {A′′_{n}} are increasing, we get

μ( ⋃_{n=N}^{∞} A_{n}) = μΛ_{l} > 0 ∀N > 0. So we can find an n > 0 with μA_{n} > 8^{m}l^{2m}K_{1}K_{2}e^{−nγ}.

On the other hand, from the following three claims we get

\begin{aligned}μA_{n} & ≤ e^{n(mλ_{−}−λ+h−(2m+3)γ)} · 8^{m}l^{2m}K_{1}K_{2}e^{n(−mλ_{−}+λ+(2m+1)γ)} · e^{−n(h−γ)} \\ & = 8^{m}l^{2m}K_{1}K_{2}e^{−nγ}. \end{aligned}

This contradiction means that (4.1) is not true.

Claim 1. The set A_{n} can be covered by e^{n(mλ_{−}−λ+h−(2m+3)γ)} balls B(x, 2ϵe^{−nλ_{−}}) with x ∈ A_{n}.

Claim 2. ∀x ∈ A_{n}, B(x, 2ϵe^{−nλ_{−}})∩A_{n} can be covered by 8^{m}l^{2m}K_{1}K_{2}e^{n(−mλ_{−}+λ+(2m+1)γ)} (n, ϵl^{−2}e^{−2nγ})-balls B_{n}(y, ϵl^{−2}e^{−2nγ}) with y ∈ A_{n}.

Claim 3. ∀y ∈ A_{n}, B_{n}(y, ϵl^{−2}e^{−2nγ}) has measure less than or equal to e^{−n(h−γ)}, i.e. μB_{n}(y, ϵl^{−2}e^{−2nγ}) ≤ e^{−n(h−γ)}.

Proof of Claim 1.

Since A_{n} ⊂ A′′′_{n}, each ϵe^{−nλ_{−}}-ball about x ∈ A_{n} has measure greater than or equal to exp[−n(mλ_{−} − λ + h − (2m + 3)γ)]. So there are at most

exp n(mλ_{−} − λ + h − (2m + 3)γ)

disjoint balls of radius ϵe^{−nλ_{−}} about points in A_{n}. The same number of balls of radius 2ϵe^{−nλ_{−}} about points in A_{n} can cover A_{n}.

Proof of Claim 2.

Since A_{n} ⊂ Λ_{l}, by Proposition 1 and the fact

\frac{ϵ}{2}l^{−2}e^{−2nγ} · Kle^{−n(λ_{s}−3γ)} = \frac{ϵ}{2}Kl^{−1}e^{−n(λ_{s}−γ)} < ϵe^{−n(λ_{s}−γ)} < ϵe^{−nλ_{−}},

we know that B_{n}(y, \frac{ϵ}{2}l^{−2}e^{−2nγ}) ⊂ B(y, ϵe^{−nλ_{−}}) ∀ y ∈ A_{n}. So, if y ∈ A_{n} ∩ B(x, 2ϵe^{−nλ_{−}}) for some x ∈ M, then B_{n}(y, \frac{ϵ}{2}l^{−2}e^{−2nγ}) ⊂ B(x, 4ϵe^{−nλ_{−}}).

Since A_{n} ⊂ A′′_{n}, Lemma 2 implies that for any y ∈ A_{n},

νB_{n}(y, \frac{ϵ}{2}l^{−2}e^{−2nγ}) ≥ K_{2}^{−1}(\frac{ϵ}{2})^{m}l^{−2m}e^{−2nmγ} · e^{−n(λ+γ)} = ϵ^{m}2^{−m}l^{−2m}K_{2}^{−1}e^{−n(λ+(2m+1)γ)}.

By Lemma 1,

νB(x, 4ϵe^{−nλ_{−}}) ≤ 4^{m}K_{1}ϵ^{m}e^{−nmλ_{−}}.

Hence, there are at most

ϵ^{−m}2^{m}l^{2m}K_{2}e^{n(λ+(2m+1)γ)} · 4^{m}K_{1}ϵ^{m}e^{−nmλ_{−}} = 8^{m}l^{2m}K_{1}K_{2}e^{n(−mλ_{−}+λ+(2m+1)γ)}

disjoint (n, \frac{ϵ}{2}l^{−2}e^{−2nγ})-balls about points in A_{n} contained in B(x, 4ϵe^{−nλ_{−}}). So the same number of (n, ϵl^{−2}e^{−2nγ})-balls can cover B(x, 2ϵe^{−nλ_{−}}).

Proof of Claim 3.

This is because A_{n} ⊂ A′_{n}.

Proof of Proposition 2 (The Second Inequality).

The proof is analogous to that for the first inequality. We only give an outline here. Write λ_{+} = λ_{1} + 2γ. Suppose for some small γ and ϵ,

lim sup_{n→∞} −\frac{1}{nλ_{+}} log μB(x, ϵe^{−nλ_{+}}) > m − \frac{λ − h − (2m + 3)γ}{λ_{+}} μ-a.e.x ∈ Λ.

Fix l ≥ 1 such that Kl^{−1} ≤ 1 and μΛ_{l} > 0. Put

A′_{n} = {x ∈ Λ : μB_{k}(x, ϵ) > exp[−k(h + γ)] ∀k ≥ n},
A′′_{n} = {x ∈ Λ : |Df_{x}^{k}| > exp k(λ − γ) ∀k ≥ n},
A′′′_{n} = {x ∈ Λ : μB(x, ϵl^{−2}e^{−n(λ_{+}+2γ)}) ≤ exp[−n(mλ_{+} − λ + h + (2m + 3)γ)] },

and put A_{n} = A′_{n} ∩ A′′_{n} ∩ A′′′_{n} ∩ Λ_{l}. We can find an n > 0 with μA_{n} > 8^{m}l^{2m}K_{1}K_{2}e^{−nγ}. However, the following three claims give that

\begin{aligned}μA_{n} & ≤ e^{n(h+γ)} · 8^{m}l^{2m}K_{1}K_{2}e^{n(mλ_{+}−λ+(2m+1)γ)} · e^{−n(mλ_{+}−λ+h+(2m+3)γ)} \\ & = 8^{m}l^{2m}K_{1}K_{2}e^{−nγ}, \end{aligned}

a contradiction.

Claim 4. The set A_{n} can be covered by e^{n(h+γ)} (n, 2ϵ)-balls B_{n}(x, 2ϵ) centered at points x ∈ A_{n}.

Claim 5. ∀x ∈ A_{n}, B_{n}(x, 2ϵ) can be covered by 8^{m}l^{2m}K_{1}K_{2}e^{n(mλ_{+}−λ+(2m+1)γ)} balls B(y, ϵl^{−2}e^{−n(λ_{+}+2γ)}) with center points y ∈ A_{n}.

Claim 6. ∀y ∈ A_{n}, the ball B(y, ϵl^{−2}e^{−n(λ_{+}+2γ)}) has measure less than or equal to e^{−n(mλ_{+}−λ+h+(2m+3)γ)}.

Proof of Claim 4.

Since A_{n} ⊂ A′_{n}, there exist at most e^{n(h+γ)} disjoint (n, ϵ)-balls B_{n}(x, ϵ) with x ∈ A_{n}. Hence the same number of (n, 2ϵ)-balls about points in A_{n} can cover A_{n}.

Proof of Claim 5.

Since A_{n} ⊂ Λ_{l}, by Proposition 1 we have B(y, \frac{ϵ}{2}l^{−2}e^{−n(λ_{+}+2γ)}) ⊂ B_{n}(y, ϵe^{−2nγ}) ⊂ B_{n}(y, ϵ) ∀y ∈ A_{n}. So if y ∈ A_{n} ∩ B_{n}(x, 2ϵ) for some x ∈ S, then B(y, \frac{ϵ}{2}l^{−2}e^{−n(λ_{+}+2γ)}) ⊂ B_{n}(x, 4ϵ).

Since A_{n} ⊂ A′′_{n}, by Lemma 2 we get νB_{n}(x, 4ϵ) ≤ K_{2}4^{m}ϵ^{m}e^{−n(λ−γ)} ∀x ∈ A_{n}. By Lemma 1 νB(y, \frac{ϵ}{2}l^{−2}e^{−n(λ_{+}+2γ)}) ≥ K_{1}^{−1}ϵ^{m}2^{−m}l^{−2m}e^{−nm(λ_{+}+2γ)}. So there are at most

8^{m}l^{2m}K_{1}K_{2}e^{n(mλ_{+}−λ+(2m+1)γ)}

disjoint balls of radius \frac{ϵ}{2}l^{−2}e^{−n(λ_{+}+2γ)} contained in B_{n}(x, 4ϵ). Therefore, B_{n}(x, 2ϵ) can be covered by the the same number of ϵl^{−2}e^{−n(λ_{+}+2γ)}-balls.

Proof of Claim 6.

This is because A_{n} ⊂ A′′′_{n}.

To prove Theorem B, we also need the following fact, a version of which is in [Y].

Fact 2. Suppose υ is a Borel probability measure on a manifold M. If there is a sequence of positive real numbers \{r_{n}\} with lim_{n→∞} r_{n} = 0 and lim_{n→∞} \frac{log r_{n+1}}{log r_{n}} = 1 such that

\underline{δ} ≤ lim inf_{n→∞} \frac{log υB(x, r_{n})}{log r_{n}} ≤ lim sup_{n→∞} \frac{log υB(x, r_{n})}{log r_{n}} ≤ \overline{δ}

for υ-a.e. x ∈ M, then

\underline{δ} ≤ D(υ) ≤ \overline{δ},

where D(υ) is dim_{H}(υ), \underline{dim}_{B}(υ), or \overline{dim}_{B}(υ).

It should be pointed out that by (2.2) and (2.4), D(υ) also can be \underline{dim}_{L}(υ) or \overline{dim}_{L}(υ).

Proof of Theorem B.

i) Take l > 0 such that μΛ_{l} > 0. By Proposition 1, for μ-a.e. x ∈ Λ_{l}, if 0 < ϵ ≤ min\{ϵ_{3}, \frac{1}{2}l^{−2}\}, then

lim inf_{n→∞} \frac{log μB(x, ϵe^{−n(λ_{1}+4γ)})}{log ϵe^{−n(λ_{1}+4γ)}} ≥ lim inf_{n→∞} −\frac{1}{n(λ_{1} + 4γ)} log μB_{n}(x, ϵKle^{−2nγ})
≥ lim inf_{n→∞} −\frac{1}{n(λ_{1} + 4γ)} log μB_{n}(x, ϵ) = \frac{h}{λ_{1} + 4γ} ,

Since γ is arbitrary, by Fact 2 we have D(μ) ≥ \frac{h}{λ_{1}} . The upper bound D(μ) ≤ \frac{h}{λ_{s}} can be

obtained similarly.

ii) This follows from Proposition 2 and Fact 2.

§5 Proof of Theorem A

Take r > 0 such that for some 1 < κ′ ≤ κ, ∥Df(y)u∥ ≥ κ′∥u∥ ∀ y ∈ B(Λ, r) ∀ u ∈ T_{y}M. This is possible because f is uniformly expanding on Λ. Proposition 3. For any ϵ < min\{r, \frac{1}{2}ϵ_{2}\},

P (f, − log |Df(x)|) = lim_{n→∞} \frac{1}{n} log νB_{n}(Λ, ϵ) ≤ 0.

Proof. Denote ϕ(x) = − log |Df(x)| and S_{n}ϕ(x) = ∑_{i=0}^{n} ϕ(f^{i}x) = − log |Df^{n}(x)|.

Take 0 < η < ϵ. Since f is expanding, there exists k > 0 such that if x ∈ Λ, y ∈ B(Λ, r) satisfy d(f^{i}x, f^{i}y) ≤ ϵ ∀0 ≤ i ≤ k, then d(x, y) ≤ η. So B_{n+k}(Λ, ϵ) ⊂ B_{n}(Λ, η) ∀ n ≥ 0.

Take an (n, η) spanning set S for Λ. We have

B_{n+k}(Λ, ϵ) ⊂ B_{n}(Λ, η) ⊂ ⋃_{x∈S} B_{n}(x, 2η).

By Lemma 2, νB_{n}(x, 2η) ≤ K_{2}2^{m}η^{m}|Df^{n}(x)|^{−1} = K_{2}2^{m}η^{m}e^{S_{n}ϕ(x)}. Thus,

νB_{n+k}(Λ, ϵ) ≤ ∑_{x∈S} νB_{n}(x, 2η) ≤ K_{2}2^{m}η^{m} ∑_{x∈S} e^{S_{n}ϕ(x)}.

So, with the same notation as in [W] §9.1, we get

νB_{n+k}(Λ, ϵ) ≤ K_{2}2^{m}η^{m}Q_{n}(f, ϕ, η),

lim sup_{n→∞} \frac{1}{n} log νB_{n}(Λ, ϵ) ≤ lim sup_{n→∞} \frac{1}{n} log Q_{n}(f, ϕ, η).

Since the left hand side is independent of η, we have

lim sup_{n→∞} \frac{1}{n} log νB_{n}(Λ, ϵ) ≤ P (f, ϕ).

The opposite inequality can be proved in an analogous way.

Before proving Theorem A, we give the following estimates.

Proposition 4. With the notations as in Theorem A,

i) \frac{h_{top}}{log ∥Df|_{Λ}∥} ≤ D(Λ) ≤ \frac{h_{top}}{log[|Df|_{Λ}|]}, and

ii) m + \frac{P}{log[|Df|_{Λ}|]} ≤ D(Λ) ≤ m + \frac{P}{log ∥Df|_{Λ}∥}.

Proof. Note that for any invariant measure μ on Λ, log ∥Df|_{Λ}∥ ≥ λ_{1} and log [|Df|_{Λ}|] ≤ λ_{s}.

Therefore the lower bound D(Λ) ≥ \frac{h_{top}}{log ∥Df|_{Λ}∥} in Part i) follows from (2.1), (2.2), Theorem

B and the fact that h_{top} = sup{h_{μ}(f) : μ ∈ E(Λ, f)}, where E(Λ, f) is the set of all ergodic measures of f on Λ.

By using the facts that P = sup{h_{μ}(f) − ∫ log |Df(x)|dμ(x) : μ ∈ E(Λ, f)} and

λ = ∫ log |Df(x)|dμ(x), we can obtain the lower bound D(Λ) ≥ m + \frac{P}{log[|Df|_{Λ}|]}.

Now we prove the upper bounds in both parts. By (2.3) we only need prove that

\overline{dim}_{B}(Λ) ≤ min\{\frac{h_{top}}{log[|Df|_{Λ}|]}, m + \frac{P}{log ∥Df|_{Λ}∥}\}.  (5.1)

For any 0 < β < [|Df|_{Λ}|] − 1, there exists ϵ_{0} > 0 such that ∀y ∈ B(Λ, ϵ_{0}), [|Df(y)|] ≥ [|Df|_{Λ}|] − β and ∥Df(y)∥ ≤ ∥Df|_{Λ}∥ + β. Denote

χ_{−} = log([|Df|_{Λ}|] − β)  and  χ_{+} = log(∥Df|_{Λ}∥ + β).  (5.2)

We assume 0 < ϵ < min\{r, ϵ_{1}, \frac{1}{2}ϵ_{2}, \frac{1}{2}ϵ_{o}\} so that Lemma 1 and Proposition 3 can be applied to this ϵ as well.

We have that ∀x ∈ B(λ, ϵ), B_{n}(x, ϵ) ⊂ B(x, ϵ([|Df|_{Λ}|] − β)^{−n}) = B(x, ϵe^{−nχ_{−}}). So, if S is an (n, ϵ) spanning set for Λ, then {B(x, ϵe^{−nχ_{−}}) : x ∈ S} is a cover of Λ. This implies that

N(ϵe^{−nχ_{−}}) ≤ N(n, ϵ),

where N(n, ϵ) is the smallest cardinality of any (n, ϵ) spanning set for Λ, and N(ϵe^{−nχ_{−}}) is the minimum number of balls of radius ϵe^{−nχ_{−}} which cover Λ. By the definitions of \overline{dim}_{B}(Λ) and topological entropy,

\overline{dim}_{B}(Λ) = lim sup_{n→0} \frac{log N(ϵe^{−nχ_{−}})}{− log ϵe^{−nχ_{−}}} ≤ lim sup_{n→0} \frac{log N(n, ϵ)}{nχ_{−}} = \frac{h_{top}}{χ_{−}} .  (5.3)

By Proposition 3, there exists N > 0 such that if n > N, then νB_{n}(Λ, ϵ) < e^{n(P +β)}. Any ϵe^{−nχ_{+}}-ball about a point in Λ must be contained in B_{n}(Λ, ϵ). Every such ϵe^{−nχ_{+}}-ball has volume larger than K_{1}^{−1}ϵ^{m}e^{−nmχ_{+}}. So B_{n}(Λ, ϵ) contains at most K_{1}ϵ^{−m}e^{n(mχ_{+}+P +β)} disjoint ϵe^{−nχ_{+}}-balls centered in Λ. This implies that

N(2ϵe^{−nχ_{+}}) ≤ K_{1}ϵ^{−m}e^{n(mχ_{+}+P +β)}.

Thus

\overline{dim}_{B}(Λ) ≤ lim sup_{n→∞} \frac{log N(2ϵe^{−nχ_{+}})}{− log 2ϵe^{−nχ_{+}}} ≤ m + \frac{P + β}{χ_{+}} .  (5.4)

Since β is arbitrary, (5.1) follows from (5.2)–(5.4). This finishes the proof.

Proof of Theorem A.

Since h(f^{n}|_{Λ}) = nh_{top} and P (f^{n}|_{Λ}, −log|Df^{n}(x)|) = nP, we can use f^{n} instead of f and then take limit to obtain Theorem A from Proposition 4.

Acknowledgement. I would like to thank Professor Lai-Sang Young, my former advisor, for many important suggestions. I would also thank the referee for carefully reading the manuscript and for suggesting various improvements.

REFERENCES

[B] Bedford, T.: Crinkly curves, Markov partitions and box dimension in self-similar sets. Ph.D. Thesis, University of Warwick

[BK] Brin, M., Katok, A.: On local entropy. In: Geometric dynamics, Lecture notes in mathematics No.1007, Berlin-New York: Springer 1983, pp. 30-38.

[BU] Bedford, T., Urbański, M.: The box and Hausdorff dimension of self-affine sets. Ergodic Theory & Dynamical Systems. 10, 627-644 (1990)

[F1] Falconer, K.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103, 339-350 (1988)

[F2] Falconer, K.: Fractal geometry – mathematical foundations and applications. New York: Wiley, 1990

[G] Gu, X.: An upper bound for the Hausdorff dimension of a hyperbolic set. Nonlinearity 4, 927-934 (1991)

[GL] Gatzouras, D., Lalley, S.: Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41, 533-568 (1992)

[H] Hutchinson, J. E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713-747 (1981)

[L] Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229-238 (1981)

[LY] Ledrappier, F., Young, L-S.: The metric entropy of diffeomorphisms. Annals of Math. 122, 509-574 (1985)

[Mc] McMullen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96, 1-9 (1984)

[MM] McCluskey, H., Manning, A.: Hausdorff dimension for horseshoe. Ergodic Theory &

Dynamical Systems 3, 251-260 (1983)

[P] Petersen, K.: Ergodic theory. Cambridge: Cambridge University Press 1983

[R] Ruelle, D.: Bowen’s formula for the Hausdorff dimension of self-similar sets. In: Scaling and self-similarity in physics, Progress in Physics 7. Frohlich, J. (ed.). Boston: Birkhäuser 1983.

[W] Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics, vol. 79. Berlin, Heidelberg, New York: Springer-Verlag 1981

[Y] Young, L.-S.: Dimension, entropy and Lyapunov exponents. Ergodic Theory & Dynamical Systems 2, 109-129 (1982)