Nonadditive Measure-theoretic Pressure and Applications to Dimensions of an Ergodic Measure

Yongluo Cao^{†}, Huyi Hu^{‡} and Yun Zhao^{†,∗}

† Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, P.R.China

(email: ylcao@suda.edu.cn, zhaoyun@suda.edu.cn)

‡ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (email: hu@math.msu.edu)

Abstract. Without any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measuretheoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact set, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the supadditive measure-theoretic pressure which has similar formalism as the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Haudorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

Key words and phrases. nonadditive, measure-theoretic pressure, variational principle, ergodic measure, Hausdorff dimension.

0 Introduction.

It is well-known that the topological pressure for additive potentials was first introduced by Ruelle for expansive maps acting on compact metric spaces ([24]), furthermore he formulated a variational principle for the topological pressure in that paper. Later, Walters ([26]) generalized these results to general continuous maps on compact metric spaces. In [23], Pesin and Pitskel’ defined the topological pressure for noncompact sets which is a generalization of Bowen’s definition of topological entropy for noncompact sets ([4]), and they proved the variational principle under some supplementary conditions. The notions of the topological pressure, variational principle and equilibrium states play a fundamental role in statistical mechanics, ergodic theory and dynamical systems (see the books [5, 27]).

Since the work of Bowen ([6]), topological pressure becomes a fundamental tool for study of the dimension theory in conformal dynamical systems (see [22]). Different versions of topological pressure were defined in dimension theory and ergodic theory. It has been found out that the dimension of nonconformal repellers can be well estimated by the zero of pressure function (see e.g. [1, 2, 12, 13, 29]). Falconer ([13]) considered the thermodynamic formalism for subadditive potentials on mixing repellers. He proved the variational principle under some Lipschitz conditions and bounded distortion assumptions on subadditive potentials. Barreira ([2]) defined topological pressure for arbitrary sequences of potentials on arbitrary subsets of compact metric spaces, and proved the variational principle under some convergence assumptions on the potentials. However, the conditions given by Falconer and Barreira are not usually satisfied by general subadditive potentials.

02000 Mathematics Subject classification: 37D35, 37A35, 37C45 ∗Corresponding author

In [8], the authors generalized the results of Ruelle and Walters to subadditive potentials in general compact dynamical systems. They defined the subadditive topological pressure and gave a variational principle for the subadditive topological pressure. We mention that their result do not need any additional conditions on either the subadditive potentials or the spaces, as long as they are compact metric spaces. In [30], the authors defined the subadditive measure-theoretic pressure by using spanning sets, and obtained a formalism similar to that for additive measure-theoretic pressure in [15] under tempered variation assumptions on subadditive potentials. Another equivalent definition of subadditive measure-theoretic pressure is given in [31] under the same conditions on the sub-additive potentials by using the definition of topological pressure on noncompact sets. In [28], Zhang studied the local measure-theoretic pressures for subadditive potentials. The pressure is local in the sense that an open cover is fixed.

Part of this paper is a continuation of the work in [30] and [31]. We modify the definition of subadditive measure-theoretic pressure there, and remove the extra condition for the formulas. More precisely, for an ergodic measure μ, the subadditive measure-theoretic pressure defined by using Carathéodory structure can be expressed as the sum of the measure-theoretic entropy and the integral of the limit of the average value of the subadditive potentials. Consequently, this paper gives equivalence to an alternative definition of subadditive measure-theoretic pressure by considering spanning sets about which the Bowen balls only cover a set of measure greater than or equal to 1 − δ. The results we get here do not need any additional assumptions on the subadditive sequences and the topological dynamical systems, except for compactness of the spaces. Meanwhile, we also define the measure-theoretic pressure for supadditive potentials in a similar way and obtain the same properties.

The present work is also motivated by the dimension theory in dynamical systems. Dimensions of a compact invariant set can often be determined or estimated by a unique root of certain pressure functions. As an application, we proved that the dimension of an average conformal repeller which is introduced in [1] satisfies a variation principle, i.e., the dimension of this repeller is equal to the dimension of some ergodic measure supported on it. Moreover, for each ergodic measure supported on an average conformal repeller, we could construct a certain set with the same dimension of the measure.

The paper is organized in the following manner. The main results, as well as definitions of the measure-theoretic pressure and lower and upper capacity measure-theoretic pressure for subaddtive potentials, are given in Section 1. We prove Theorem A and B, the results related to subadditive potentials, in Section 2. Section 3 is devoted to supadditive potentials, where we give definitions of the pressures, and prove Theorem C and D. Section 4 is for application to dimension of average conformal repellers, where the results are stated in Theorem E and F.

1 Main results

Let (X, T) be a topological dynamical systems(TDS), that is, X is a compact metric space with a metric d, and T : X → X is a continuous transformation. Denote by M_{T} and E_{T} the set of all T -invariant Borel probability measures on X and the set of ergodic measures respectively. For each μ ∈ M_{T}, let h_{μ}(T) denote the measure-theoretic entropy of T with respect to μ.

A sequence F = {f_{n}}_{n≥1} of continuous functions on X is a subadditive potential on X, if

f_{n+m}(x) ≤ f_{n}(x) + f_{m}(T^{n}x) for all x ∈ X, n, m ∈ ℕ.

For μ ∈ M_{T}, let F_{∗}(μ) denote the following limit

F_{∗}(μ) = lim_{n→∞} \frac{1}{n} ∫ f_{n}dμ = inf_{n≥1}\{\frac{1}{n} ∫ f_{n}dμ\}.

The limit exists since {∫ f_{n}dμ}_{n≥1} is a subadditive sequence. Also, by subadditive ergodic theorem [18] the limit lim_{n→∞}(1/n)f_{n} exists μ-almost everywhere for any μ ∈ M_{T}.

Let d_{n}(x, y) = max{d(T^{i}(x), T^{i}(y)) : i = 0, ⋯, n − 1} for any x, y ∈ X, and B_{n}(x, ϵ) = {y ∈ X : d_{n}(x, y) < ϵ}. A set E ⊆ X is said to be an (n, ϵ)-separated subset of X with respect to T if x, y ∈ E, x ≠ y, implies d_{n}(x, y) > ϵ. A set F ⊆ X is said to be an (n, ϵ)-spanning subset of X with respect to T if ∀x ∈ X, ∃y ∈ F with d_{n}(x, y) ≤ ϵ. For each μ ∈ M_{T}, 0 < δ < 1, n ≥ 1 and ϵ > 0, a subset F ⊆ X is an (n, ϵ, δ)-spanning set if the union ⋃_{x∈F} B_{n}(x, ϵ) has μ-measure more than or equal to 1 − δ.

Recall that the subadditive topological pressure of T with respect to a subadditive potential F = {f_{n}}_{n≥1} is give by

P (T, F) = lim_{ϵ→0} P (T, F, ϵ),

where

P (T, F, ϵ) = lim sup_{n→∞} \frac{1}{n} log P_{n}(T, F, ϵ),

P_{n}(T, F, ϵ) = sup{∑_{x∈E} e^{f_{n}(x)} : E is an (n, ϵ)-separated subset of X}.

(See e.g. [2], [8].) It satisfies a variational principle (see [8] for a proof and [32] for its random version).

In [17], Katok showed that measure-theoretic entropy can be regarded as the growth rate of the minimal number of ϵ-balls in the d_{n} metric that cover a set of measure more than or equal to 1 − δ. Motivated by the observation, the following definition can be given.

Definition 1.1. Given a subadditive potential F = {f_{n}}, for μ ∈ E_{T}, 0 < δ < 1, n ≥ 1, and ϵ > 0, put

P_{μ}(T, F, n, ϵ, δ) = inf{∑_{x∈F} exp[ sup_{y∈B_{n}(x,ϵ)} f_{n}(y)] | F is an (n, ϵ, δ) − spanning set},

P_{μ}(T, F, ϵ, δ) = lim sup_{n→∞} \frac{1}{n} log P_{μ}(T, F, n, ϵ, δ),

P_{μ}(T, F, δ) = lim inf_{ϵ→0} P_{μ}(T, F, ϵ, δ),

P_{μ}(T, F) = lim_{δ→0} P_{μ}(T, F, δ).

P_{μ}(T, F) is said to be the subadditive measure-theoretic pressure of T with respect to F.

Remark 1.1. It is easy to see that P_{μ}(T, F, δ) increases with δ. So the limit in the last formula exists. In fact, it is proved in [11, Theorem 2.3] that P_{μ}(T, F, δ) is independent of δ. Hence, the limit of δ → 0 is redundant in the definition. The same phenomenon can also be seen for measure-theoretic entropy (see [17, Theorem 1.1]).

Remark 1.2. If F = {f_{n}} is additive generated by a continuous function φ, that is, f_{n}(x) = ∑_{i=0}^{n−1} φ(T^{i}x) for some continuous function φ : X → ℝ, then we simply write P_{μ}(T, F) as P_{μ}(T, φ).

An alternative definition of subadditive measure-theoretic pressure can be given by using the theory of Carathéodory structure (see [22] for more details in additive case).

Let Z ⊆ X be a subset of X, which does not have to be compact nor T -invariant. Fix ϵ > 0, we call Γ = {B_{n_{i}}(x_{i}, ϵ)}_{i} a cover of Z if Z ⊆ ⋃_{i} B_{n_{i}}(x_{i}, ϵ). For Γ = {B_{n_{i}}(x_{i}, ϵ)}_{i}, set n(Γ) = min_{i}{n_{i}}.

The theory of Carathéodory dimension characteristic ensures the following definitions.

Definition 1.2. Let s ≥ 0, put

M(Z, F, s, N, ϵ) = inf_{Γ} ∑_{i} exp(−sn_{i} + sup_{y∈B_{n_{i}} (x_{i},ϵ)} f_{n_{i}}(y)),  (1.1)

where the infimum is taken over all covers Γ of Z with n(Γ) ≥ N. Then let

m(Z, F, s, ϵ) = lim_{N→∞} M(Z, F, s, N, ϵ),  (1.2)
P_{Z}(T, F, ϵ) = inf{s : m(Z, F, s, ϵ) = 0} = sup{s : m(Z, F, s, ϵ) = +∞},  (1.3)
P_{Z}(T, F) = lim inf_{ϵ→0} P_{Z}(T, F, ϵ),  (1.4)

where P_{Z}(T, F) is called a subadditive topological pressure of T on the set Z (w.r.t. F). Further, for μ ∈ M_{T}, put

P_{μ}^{∗}(T, F, ϵ) = inf{P_{Z}(T, F, ϵ) : μ(Z) = 1},
P_{μ}^{∗}(T, F) = lim inf_{ϵ→0} P_{μ}^{∗}(T, F, ϵ),  (1.5)

where P_{μ}^{∗}(T, F) is called a subadditive measure-theoretic pressure of T with respect to μ.

It is easy to see that the definition is consistent with that given in [2] by using arbitrary open covers.

Lower and upper capacity topological pressure for additive sequence were defined in [22]. Now we give similar definitions:

Definition 1.3. Put

Λ(Z, F, N, ϵ) = inf_{Γ} ∑_{i} exp( sup_{y∈B_{N} (x_{i},ϵ)} f_{N}(y)),

where the infimum is taken over all covers Γ of Z with n_{i} = N for all i. Then we set

\underline{CP}_{Z}(T, F, ϵ) = lim inf_{N→∞} \frac{1}{N} log Λ(Z, F, N, ϵ),  (1.6)
\overline{CP}_{Z}(T, F, ϵ) = lim sup_{N→∞} \frac{1}{N} log Λ(Z, F, N, ϵ).  (1.7)

For μ ∈ M_{T}, define

\underline{CP} _{μ}^{∗}(T, F, ϵ) = lim_{δ→0} inf\{\underline{CP}_{Z}(T, F, ϵ) : μ(Z) ≥ 1 − δ\},

\overline{CP} _{μ}^{∗}(T, F, ϵ) = lim_{δ→0} inf\{\overline{CP}_{Z}(T, F, ϵ) : μ(Z) ≥ 1 − δ\}.

The subadditive lower and upper capacity measure-theoretic pressure of T with respect to measure μ are defined by

\underline{CP} _{μ}^{∗}(T, F) = lim inf_{ϵ→0} \underline{CP} _{μ}^{∗}(T, F, ϵ),  (1.8)
\overline{CP} _{μ}^{∗}(T, F) = lim inf_{ϵ→0} \overline{CP} _{μ}^{∗}(T, F, ϵ).  (1.9)

Theorem A. Let (X, T) be a TDS and F = {f_{n}}_{n≥1} a subadditive potential on X. For any μ ∈ E_{T} with F_{∗}(μ) ≠ −∞, we have

P_{μ}^{∗}(T, F) = \underline{CP} _{μ}^{∗}(T, F) = \overline{CP} _{μ}^{∗}(T, F) = P_{μ}(T, F) = h_{μ}(T) + F_{∗}(μ).

Remark 1.3. The results still apply for F_{∗}(μ) = −∞ if h_{μ}(T) < ∞.

Remark 1.4. If F = {f_{n}} is an additive sequence generated by a continuous function φ : X → ℝ, then we have P_{μ}(T, φ) = h_{μ}(T) + ∫ φdμ, So Theorem A extends the results in [22] and [15] to the subadditive case. Also, the last equality was proved in [11].

Remark 1.5. By the definition of P_{μ}^{∗}(T, F, ϵ) and P_{μ}^{∗}(T, F), the theorem gives h_{μ}(T) + F_{∗}(μ) = inf{P_{Z}(T, F) : μ(Z) = 1}, as in [22]. We call it the inverse variational principle.

The next theorem says that the infimum in the inverse variational principle can be attained on certain sets.

Theorem B. Let (X, T) be TDS and F = {f_{n}} a subadditive potential on X. For any μ ∈ E_{T} with F_{∗}(μ) ≠ −∞, let

K = \{x ∈ X : lim_{ϵ→0} lim sup_{n→∞} \frac{− log μ(B_{n}(x, ϵ))}{n} = h_{μ}(T) and lim_{n→∞} \frac{1}{n}f_{n}(x) = F_{∗}(μ)\} .

Then we have

P_{μ}(T, F) = P_{K}(T, F) = \underline{CP}_{K}(T, F) = \overline{CP}_{K}(T, F).

Similar to subadditive sequences, we can study supadditive sequences. A sequence Φ = {ϕ_{n}}_{n≥1} of continuous functions on X is a supadditive potentials on X, if

ϕ_{n+m}(x) ≥ ϕ_{n}(x) + ϕ_{m}(T^{n}x) for all x ∈ X, n, m ∈ ℕ.  (1.10)

Note that if Φ = {ϕ_{n}}_{n≥1} is a supadditive sequences, then −Φ = {−ϕ_{n}}_{n≥1} is a susadditive sequences. So the limit lim_{n→∞}(1/n)ϕ_{n} exists μ-almost everywhere for any μ ∈ M_{T}. For μ ∈ M_{T}, let

Φ_{∗}(μ) = lim_{n→∞} \frac{1}{n} ∫ ϕ_{n}dμ = sup_{n≥1} \{\frac{1}{n} ∫ ϕ_{n}dμ\} .

It is always bounded below by ∫ ϕ_{1}dμ.

With the sequences, we can define supadditive measure-theoretic pressure P_{μ}(T, Φ) and other quantities P_{μ}^{∗}(T, Φ), \underline{CP} _{μ}^{∗}(T, Φ) and \overline{CP} _{μ}^{∗}(T, Φ), etc. (see Section 3 for precise definitions.)

In [1], the authors gave the variational principle for supadditive topological pressure for C^{1} average conformal expanding maps T where the potentials are of the form {ϕ_{n}(x)} = {−t log ∥DT^{n}(x)∥} with t > 0. However, it is still open whether variational principle holds for supadditive topological pressure for a general TDS. Here we show that the supadditive measure-theoretic pressure has similar formalisms as subadditive measure-theoretic pressure.

Theorem C. Let (X, T) be a TDS and Φ = {ϕ_{n}} a supadditive potential on X. For any μ ∈ E_{T}, we have

P_{μ}^{∗}(T, Φ) = \underline{CP} _{μ}^{∗}(T, Φ) = \overline{CP} _{μ}^{∗}(T, Φ) = P_{μ}(T, Φ) = h_{μ}(T) + Φ_{∗}(μ).

Remark 1.6. For each μ ∈ E_{T}, from the definition of P_{μ}^{∗}(T, Φ, ϵ) and P_{μ}^{∗}(T, Φ) given in Section 3, the theorem gives the inverse variational principle h_{μ}(T) + Φ_{∗}(μ) = inf{P_{Z}(T, Φ) : μ(Z) = 1}.

Theorem D. Let (X, T) be TDS, and Φ = {ϕ_{n}} a supadditive potential on X. For any μ ∈ E_{T}, let

K = \{x ∈ X : lim_{ϵ→0} lim sup_{n→∞} \frac{− log μ(B_{n}(x, ϵ))}{n} = h_{μ}(T) and lim_{n→∞} \frac{1}{n}ϕ_{n}(x) = Φ_{∗}(μ)\}.

Then we have

P_{μ}(T, Φ) = P_{K}(T, Φ) = \underline{CP}_{K}(T, Φ) = \overline{CP}_{K}(T, Φ).

2 Subadditive measure-theoretic pressures

We start with the section by some properties of pressures for subadditive potentials.

Proposition 2.1. Let (X, T) be a TDS and F = {f_{n}} a subadditive potential. Then the following properties hold:

(i) P_{Z_{1}}(T, F) ≤ P_{Z_{2}}(T, F) if Z_{1} ⊂ Z_{2}, where P is P, \underline{CP} or \overline{CP};

(ii) P_{Z}(T, F) = sup_{i≥1} P_{Z_{i}}(T, F) and P_{Z}(T, F) ≥ sup_{i≥1} P_{Z_{i}}(T, F), where Z = ⋃_{i≥1} Z_{i}, and P is \underline{CP} or \overline{CP};

(iii) P_{Z}(T, F) ≤ \underline{CP}_{Z}(T, F) ≤ \overline{CP}_{Z}(T, F) for any subset Z ⊂ X;

(iv) P_{μ}^{∗}(T, F, ϵ) ≤ \underline{CP} _{μ}^{∗}(T, F, ϵ) ≤ \overline{CP} _{μ}^{∗}(T, F, ϵ), and P_{μ}^{∗}(T, F) ≤ \underline{CP} _{μ}^{∗}(T, F) ≤ \overline{CP} _{μ}^{∗}(T, F).

Proof. (i) and (ii) are dirctly follow from the definition. And (iii) is immediately from similarly arguments as in [2, Theorem 1.4 (a)]. (iv) follows from (iii) immediately by the definition.

Recall the Brin-Katok’s theorem for local entropy (see [7]), which says that if μ ∈ M_{T}, then for μ-almost every x ∈ X,

h_{μ}(x, T) = lim_{ϵ→0} lim inf_{n→∞} −\frac{1}{n} log μ(B_{n}(x, ϵ)) = lim_{ϵ→0} lim sup_{n→∞} −\frac{1}{n} log μ(B_{n}(x, ϵ)).  (2.1)

Moreover, if μ ∈ E_{T}, then for μ−almost every x ∈ X, h_{μ}(x, T) = h_{μ}(T), and for each ϵ > 0, the following two limits are constants almost everywhere:

lim inf_{n→∞} −\frac{1}{n} log μ(B_{n}(x, ϵ)), lim sup_{n→∞} −\frac{1}{n} log μ(B_{n}(x, ϵ)).

Proof of Theorem A. The last equality is a direct consequence of Theorem 2.3 in [11]. By Proposition 2.1, we only need to prove \overline{CP} _{μ}^{∗}(T, F) ≤ h_{μ}(T) + F_{∗}(μ) and P_{μ}^{∗}(T, F) ≥ h_{μ}(T) + F_{∗}(μ).

For μ ∈ E_{T}, we first assume h_{μ}(T) is finite and set h = h_{μ}(T) ≥ 0.

Take δ > 0. Fix a positive integer k and a small number η > 0.

Take ϵ_{η} > 0 such that if ϵ ∈ (0, ϵ_{η}], then for μ-almost every x ∈ X,

h − η/2 ≤ lim inf_{n→∞} −\frac{1}{n} log μ(B_{n}(x, ϵ)) ≤ lim sup_{n→∞} −\frac{1}{n} log μ(B_{n}(x, ϵ)) ≤ h + η/2.

This is possible because of (2.1). Take 0 < ϵ ≤ min{ϵ_{η}, ϵ_{0}}, where ϵ_{0} is given in Lemma 2.2. Hence, for μ-almost every x ∈ X, there exists a number N_{1}(x) > 0 such that for any n ≥ N_{1}(x),

|\frac{1}{n} log μ(B_{n}(x, ϵ/2)) + h| ≤ η.  (2.2)

By the Birkhoff ergodic theorem, for μ-almost every x ∈ X, there exists a number N_{2}(x) > 0 such that for any n ≥ N_{2}(x),

|\frac{1}{n} ∑_{i=0}^{n−1} \frac{1}{k} f_{k}(T^{i}x) − ∫ \frac{1}{k} f_{k}dμ| ≤ η.  (2.3)

Given N > 0, set K_{N} = {x ∈ X : N_{1}(x), N_{2}(x) ≤ N}. We have that K_{N} ⊂ K_{N+1}, and ∪_{N≥0}K_{N} is a set of full measure. Therefore, one can find N_{0} > 0 for which μ(K_{N_{0}}) > 1 − δ. Fix a number N > N_{0}. Then for any n > N_{0} and any point x ∈ K_{N}, by (2.3) and Lemma 2.2 below we have

sup_{y∈B_{n}(x,ϵ)} f_{n}(y) ≤ n ∫ \frac{1}{k} f_{k}dμ + 2nη + C,

where C is a constant given in Lemma 2.2.

Let E be a maximal (n, ϵ)-separated subset of K_{N}, then K_{N} ⊆ ∪_{x∈E}B_{n}(x, ϵ). Furthermore, the balls {B_{n}(x, ϵ/2) : x ∈ E} are pairwise disjoint and by (2.2) the cardinality of E is less than or equal to exp n(h + η). Therefore, we have

Λ(K_{N}, F, n, ϵ) ≤ ∑_{x∈E} exp( sup_{y∈B_{n}(x,ϵ)} f_{n}(y))
≤ exp n(h + η) · exp [n(∫ \frac{1}{k} f_{k}dμ + 2η) + C]
= exp [n(∫ \frac{1}{k} f_{k}dμ + h + 3η) + C] .

From (1.7), we have

\overline{CP}_{K_{N}} (T, F, ϵ) ≤ ∫ \frac{1}{k} f_{k}dμ + h + 3η.

Since μ(K_{N}) ≥ 1 − δ, we have

\overline{CP} _{μ}^{∗}(T, F, ϵ) ≤ ∫ \frac{1}{k} f_{k}dμ + h + 3η.

Let ϵ → 0 and k → ∞ in the inequality, and by the arbitrariness of η, we get \overline{CP} _{μ}^{∗}(T, F) ≤ h + F_{∗}(μ).

To prove the other inequality, it is sufficient to prove that P_{Z}(T, F) ≥ h + F_{∗}(μ) for any subset Z ⊆ X of full μ-measure.

Take η > 0 and δ ∈ (0, 1/2), and denote λ = h + F_{∗}(μ) − 2η.

Let

K = \{x ∈ X : lim_{ϵ→0} lim sup_{n→∞} \frac{− log μ(B_{n}(x, ϵ))}{n} = h_{μ}(T) and lim_{n→∞} \frac{1}{n}f_{n}(x) = F_{∗}(μ)\} .

Put K′ = K ∩Z. By the Brin-Katok’s theorem for local entropy and the subadditive ergodic theorem, we have μ(K) = 1 and then μ(K′) = 1. For ϵ ∈ (0, ϵ_{η}], there exists a set K_{1} ⊂ K′ with μ(K_{1}) > 1 − \frac{δ}{2} and N_{1} > 0 such that for any x ∈ K_{1} and n ≥ N_{1}, we have

μ(B_{n}(x, 2ϵ)) ≤ exp(−n(h − η)).

By the subadditive ergodic theorem, there exists a set K_{2} ⊂ K′ with μ(K_{2}) > 1 − \frac{δ}{2} and N_{2} > 0 such that for any x ∈ K_{2} and n ≥ N_{2}, we have

|\frac{1}{n}f_{n}(x) − F_{∗}(μ)| < η.

Put K̃ = K_{1} ∩ K_{2} ⊂ K′, N ≥ max{N_{1}, N_{2}}. Clearly μ(K̃) > 1 − δ. We may assume further that K̃ is compact since otherwise we can approximate it from within by a compact subset. Take an open cover Γ = {B_{n_{i}}(x_{i}, ϵ)}_{i} of K̃ with n(Γ) ≥ N. Since K̃ is compact, we may assume that the cover is finite and consists of B_{n_{1}}(x_{1}, ϵ), ⋯, B_{n_{l}}(x_{l}, ϵ).

For each i = 1, ⋯, l, we choose y_{i} ∈ K̃ ∩ B_{n_{i}}(x_{i}, ϵ). Hence, B_{n_{i}}(x_{i}, ϵ) ⊂ B_{n_{i}}(y_{i}, 2ϵ), and {B_{n_{i}}(y_{i}, 2ϵ)}_{i} form a cover of K̃ as well. Now we have

∑_{B_{n_{i}} (x_{i},ϵ)∈Γ} exp(−n_{i}λ + sup_{y∈B_{n_{i}} (x_{i},ϵ)} f_{n_{i}}(y)) ≥ ∑_{i=1}^{l} exp(−n_{i}λ + f_{n_{i}}(y_{i}))

≥ ∑_{i=1}^{l} exp(−n_{i}λ + n_{i}(F_{∗}(μ) − η)) = ∑_{i=1}^{l} exp(−n_{i}(h − η)) ≥ ∑_{i=1}^{l} μ(B_{n_{i}}(y_{i}, 2ϵ)) ≥ 1 − δ ≥ \frac{1}{2}.

Note that the inequality holds for any cover Γ = {B_{n_{i}}(x_{i}, ϵ)}_{i} of K̃. Hence

M(K̃, F, λ, N, ϵ) ≥ \frac{1}{2}.

Thus m(K̃, F, λ, ϵ) ≥ 1/2. It means that

PK̃(T, F, ϵ) ≥ λ = h + F_{∗}(μ) − 2η, and PK̃(T, F) ≥ h + F_{∗}(μ) − 2η.

Using Proposition 2.1 and arbitrariness of η, we have

P_{Z}(T, F) ≥ PK̃(T, F) ≥ h + F_{∗}(μ).  (2.4)

So by definition we get P_{μ}^{∗}(T, F) ≥ h + F_{∗}(μ).

When h_{μ}(T) = +∞, modify subtly the proof of the second inequality, we can easily have P_{μ}^{∗}(T, F) = +∞. Thus we finish the proof of the theorem.

Proof of Theorem B. As in the proof of Theorem A, set h = h_{μ}(f) ≥ 0. Fix an integer k > 0 and a small real number η > 0.

For μ-almost every x ∈ X, there exists a number N_{1}(x) > 0 such that for any n ≥ N_{1}(x),

|\frac{1}{n} ∑_{i=0}^{n−1} \frac{1}{k} f_{k}(T^{i}x) − ∫ \frac{1}{k} f_{k}dμ| ≤ η.

For some ϵ ∈ (0, ϵ_{η}], where ϵ_{η} is chosen in the same way as that in the proof of Theorem A, for μ-almost every x ∈ X there exists a number N_{2}(x) > 0 such that for any n ≥ N_{2}(x),

|\frac{1}{n} log μ(B_{n}(x, ϵ/2)) + h| ≤ η.

Given N > 0, set K_{N} = {x ∈ K : N_{1}(x), N_{2}(x) ≤ N}, we have K_{N} ⊂ K_{N+1}, and ∪_{N≥0}K_{N} = K. Similarly, given δ > 0, we can find N_{0} > 0 for which μ(K_{N_{0}}) > 1 − δ.

Fix a number N ≥ N_{0}, as in the proof of Theorem A we get that \overline{CP}_{K_{N}} (T, F, ϵ) ≤ ∫ \frac{1}{k} f_{k}dμ+h+3η. Letting k → ∞, ϵ → 0, since η is arbitrary, we get

\overline{CP}_{K_{N}} (T, F) ≤ h + F_{∗}(μ).

Letting N → ∞, we have that \overline{CP}_{K}(T, F) ≤ h + F_{∗}(μ).

The inequality P_{K}(T, F) ≥ h + F_{∗}(μ) is contained in (2.4) since Z is an arbitrary set with full measure, and here we have μ(K) = 1.

By Theorem A and Proposition 2.1, we get the desired result.

Lemma 2.2. Let (X, T) be a TDS and F = {f_{n}} a subadditive potential. Fix any positive integer k, then for any η > 0, there exist an ϵ_{0} > 0 such that for any 0 < ϵ < ϵ_{0} we have

sup_{y∈B_{n}(x,ϵ)} f_{n}(y) ≤ ∑_{i=0}^{n−1} \frac{1}{k} f_{k}(T^{i}x) + nη + C,

where C = C_{k} is a constant independent of η and ϵ.

Proof. Fix a positive integer k, \frac{1}{k} f_{k}(x) is a continuous function. Hence, for any η > 0, there exist an ϵ_{0} > 0 such that for any 0 < ϵ < ϵ_{0},

d(x, y) < ϵ ⇒ d (\frac{1}{k} f_{k}(x), \frac{1}{k} f_{k}(y)) < η.  (2.5)

For each n, we rewrite n as n = sk + l, where s ≥ 0, 0 ≤ l < k. Then for any integer 0 ≤ j < k, we have

f_{n}(x) ≤ f_{j}(x) + f_{k}(T^{j}x) + ⋯ + f_{k}(T^{(s−2)k}T^{j}x) + f_{k+l−j}(T^{(s−1)k}T^{j}x),

where we take f_{0}(x) ≡ 0. Let C_{1} = max_{j=1,···2k} max_{x∈X} |f_{j}(x)|. Summing over j from 0 to k − 1, we have

kf_{n}(x) ≤ 2kC_{1} + ∑_{i=0}^{(s−1)k−1} f_{k}(T^{i}x).

Hence

f_{n}(x) ≤ 2C_{1} + ∑_{i=0}^{(s−1)k−1} \frac{1}{k} f_{k}(T^{i}x) ≤ 4C_{1} + ∑_{i=0}^{n−1} \frac{1}{k} f_{k}(T^{i}x).  (2.6)

Set C = 4C_{1}. By (2.5) we have that

sup_{y∈B_{n}(x,ϵ)} f_{n}(y) ≤ sup_{y∈B_{n}(x,ϵ)}(C + ∑_{i=0}^{n−1} \frac{1}{k} f_{k}(T^{i}y)) ≤ ∑_{i=0}^{n−1} \frac{1}{k} f_{k}(T^{i}x) + nη + C.

This completes the proof of the lemma.

3 Supadditive measure-theoretic pressures

Recall that supadditive sequence is defined in (1.10)

Definition 3.1. Let Φ = {ϕ_{n}} be a given supadditive potential. For μ ∈ E_{T}, 0 < δ < 1, n ≥ 1, and ϵ > 0, put

P_{μ}(T, Φ, n, ϵ, δ) = inf{∑_{x∈F} e^{ϕ_{n}(x)} | F is an (n, ϵ, δ)-spanning set},

P_{μ}(T, Φ, ϵ, δ) = lim sup_{n→∞} \frac{1}{n} log P_{μ}(T, Φ, n, ϵ, δ),

P_{μ}(T, Φ, δ) = lim_{ϵ→0} P_{μ}(T, Φ, ϵ, δ),

P_{μ}(T, Φ) = lim_{δ→0} P_{μ}(T, Φ, δ),

where P_{μ}(T, Φ) is called the supadditive measure-theoretic pressure of T with respect to Φ.

Note that there is small difference between the definitions of P_{μ}(T, F, n, ϵ, δ) and P_{μ}(T, Φ, n, ϵ, δ). This difference makes it possible to remove the tempered variation assumptions on the potentials as in [30, 31].

Remark 3.1. Similar to Remark 1.1, P_{μ}(T, Φ, δ) also increases with δ and therefore the limit in the last formula exists. Moreover, by Proposition 3.2 below P_{μ}(T, Φ, δ) is independent of δ.

Recall that if Γ = {B_{n_{i}}(x_{i}, ϵ)}_{i} is a cover of a subset Z ⊆ X, where ϵ > 0, then n(Γ) = min_{i}{n_{i}}.

Definition 3.2. Let Φ = {ϕ_{n}} be a given supadditive potential. For s ≥ 0, define

M(Z, Φ, s, N, ϵ) = inf_{Γ} ∑_{i} exp(−sn_{i} + ϕ_{n_{i}}(x_{i})),

where the infimum is taken over all covers Γ = {B_{n_{i}}(x_{i}, ϵ)}_{i} of Z with n(Γ) ≥ N. Then let

m(Z, Φ, s, ϵ) = lim_{N→∞} M(Z, Φ, s, N, ϵ),  (3.1)
P_{Z}(T, Φ, ϵ) = inf{s : m(Z, Φ, s, ϵ) = 0} = sup{s : m(Z, Φ, s, ϵ) = +∞},  (3.2)
P_{Z}(T, Φ) = lim inf_{ϵ→0} P_{Z}(T, Φ, ϵ),  (3.3)

where P_{Z}(T, Φ) is called a supadditive topological pressure of T on the set Z (w.r.t. Φ). Further, for μ ∈ M_{T}, put

P_{μ}^{∗}(T, Φ, ϵ) = inf{P_{Z}(T, Φ, ϵ) : μ(Z) = 1},
P_{μ}^{∗}(T, Φ) = lim inf_{ϵ→0} P_{μ}^{∗}(T, Φ, ϵ),  (3.4)

where P_{μ}^{∗}(T, Φ) is called a supadditive measure-theoretic pressure of T with respect to μ.

Definition 3.3. Put

Λ(Z, Φ, N, ϵ) = inf_{Γ} ∑_{B_{N} (x,ϵ)∈Γ} exp(ϕ_{N}(x)),

where the infimum is taken over all covers Γ of Z with n_{i} = N for all i. And then we set

\underline{CP}_{Z}(T, Φ, ϵ) = lim inf_{N→∞} \frac{1}{N} log Λ(Z, Φ, N, ϵ),  (3.5)
\overline{CP}_{Z}(T, Φ, ϵ) = lim sup_{N→∞} \frac{1}{N} log Λ(Z, Φ, N, ϵ).  (3.6)

For μ ∈ M_{T}, define

\underline{CP} _{μ}^{∗}(T, Φ, ϵ) = lim_{δ→0} inf\{\underline{CP}_{Z}(T, Φ, ϵ) : μ(Z) ≥ 1 − δ\},

\overline{CP} _{μ}^{∗}(T, Φ, ϵ) = lim_{δ→0} inf\{\overline{CP}_{Z}(T, Φ, ϵ) : μ(Z) ≥ 1 − δ\}.

The supadditive lower and upper capacity measure-theoretic pressure of T with respect to measure μ are defined by

\underline{CP} _{μ}^{∗}(T, Φ) = lim inf_{ϵ→0} \underline{CP} _{μ}^{∗}(T, Φ, ϵ),  (3.7)
\overline{CP} _{μ}^{∗}(T, Φ) = lim inf_{ϵ→0} \overline{CP} _{μ}^{∗}(T, Φ, ϵ).  (3.8)

It is easy to see that all the pressures of supadditive potentials defined as above have the same properties of the corresponding pressures of subadditive potentials. We state it here, whose proof is similar and left to the reader.

Proposition 3.1. All the properties stated in Proposition 2.1 are true if we replace the subadditive potential F by a supadditive potential Φ.

Proof of Theorem C. The last equality is proved in Proposition 3.2 below. So by Proposition 3.1, we only need to prove \overline{CP} _{μ}^{∗}(T, Φ) ≤ h_{μ}(T) + F_{∗}(μ) and P_{μ}^{∗}(T, Φ) ≥ h_{μ}(T) + F_{∗}(μ).

For μ ∈ E_{T}, we first assume h_{μ}(T) is finite and set h = h_{μ}(T) ≥ 0.

Fix a small number η > 0 and an ϵ ∈ (0, ϵ_{η}], where ϵ_{η} is determined in a similar way as in the proof of Theorem A. Hence, by the Brin-Katok theorem (see [7]) for local entropy, for μ-almost every x ∈ X there exists a number N_{1}(x) > 0 such that for any n ≥ N_{1}(x), we have

|\frac{1}{n} log μ(B_{n}(x, ϵ/2)) + h| ≤ η.  (3.9)

Since −Φ = {−ϕ_{n}} is a subadditive sequence, by the subadditive ergodic theorem, for μ-almost every x ∈ X there exists a number N_{2}(x) > 0 such that for any n ≥ N_{2}(x), we have

|\frac{1}{n}ϕ_{n}(x) − Φ_{∗}(μ)| ≤ η  (3.10)

Given N > 0, set K_{N} = {x ∈ X : N_{1}(x), N_{2}(x) ≤ N}. We have that K_{N} ⊂ K_{N+1}, and ∪_{N≥0}K_{N} is a set of full measure. Therefore, given δ > 0, we can find N_{0} > 0 for which μ(K_{N_{0}}) > 1 − δ.

Fix a number N ≥ N_{0}. Let E be a maximal (n, ϵ)-separated subset of K_{N}, then K_{N} ⊆ ∪_{x∈E}B_{n}(x, ϵ). Furthermore, the balls {B_{n}(x, ϵ/2) : x ∈ E} are pairwise disjoint. By (3.9) the cardinality of E is less than or equal to exp n(h + η). Therefore, by (3.10) we have

Λ(K_{N}, Φ, n, ϵ) ≤ ∑_{x∈E} exp(ϕ_{n}(x)) ≤ exp n(h + η) · exp n(Φ_{∗}(μ) + η)
= exp n(Φ_{∗}(μ) + h + 2η).

It follows

\overline{CP}_{K_{N}} (T, Φ, ϵ) ≤ Φ_{∗}(μ) + h + 2η.

Since μ(K_{N}) ≥ 1 − δ and μ(K_{N}) → 1 as N → ∞, we have

\overline{CP} _{μ}^{∗}(T, Φ, ϵ) ≤ Φ_{∗}(μ) + h + 2η.

Let ϵ → 0 and take limit. By arbitrariness of η, we have \overline{CP} _{μ}^{∗}(T, Φ) ≤ h + Φ_{∗}(μ).

To prove the other inequality, it is sufficient to prove that P_{Z}(T, Φ) ≥ h + Φ_{∗}(μ) for any subset Z ⊆ X of full μ-measure. Fix a subset Z with full measure and a positive integer k. Note that for

an additive sequence F = \{f_{n}\} = \{∑_{i=1}^{n} \frac{1}{k} ϕ_{k} ∘ T^{i−1}\}, P_{Z}(T, F) becomes the pressure P_{Z}(T, f_{1}) of

f_{1} = (1/k)ϕ_{k}. So if we apply the same arguments as in the proof of Theorem A for the sequence, we get the inequality as in (2.4), that is,

P_{Z}(T, \frac{1}{k} ϕ_{k}) ≥ h + ∫ \frac{1}{k} ϕ_{k}dμ.

Using lemma 3.3 below, we have

P_{Z}(T, Φ) ≥ h + ∫ \frac{1}{k} ϕ_{k}dμ.

The arbitrariness of k implies that P_{Z}(T, Φ) ≥ h + Φ_{∗}(μ).

If h_{μ}(T) = +∞, we can easily have P_{μ}^{∗}(T, Φ) = +∞. Thus we finish the proof of the theorem.

Proof of Theorem D. As in the proof of Theorem C, set h = h_{μ}(f) ≥ 0. By the same arguments, there is a sequence of subsets {K_{N}}_{N≥1} such that K_{N} ⊂ K_{N+1} and ∪_{N≥0}K_{N} = K. Moreover,

\overline{CP}_{K_{N}} (T, Φ) ≤ h + Φ_{∗}(μ)

for all sufficiently large N. Letting N → ∞, we get that \overline{CP}_{K}(T, Φ) ≤ h + Φ_{∗}(μ).

By Theorem C, the reverse inequality P_{K}(T, Φ) ≥ h + Φ_{∗}(μ) is immediate since μ(K) = 1. Hence Theorem C and Proposition 3.1 implies the desired results.

Proposition 3.2. Let (X, T) be a TDS, and Φ = {ϕ_{n}} a supadditive potential on X. For μ ∈ E_{T}, we have

P_{μ}(T, Φ) = lim_{ϵ→0} lim inf_{n→∞} \frac{1}{n} log P_{μ}(T, Φ, n, ϵ, δ) = h_{μ}(T) + Φ_{∗}(μ).

Proof. Fix a positive integer k and a small number δ > 0. Take η > 0. Let ϵ_{0} be as in Sublemma 3.5. Then by Lemma 3.4, for any ϵ ∈ (0, ϵ_{0}], n > 0, we can get

P_{μ}(T, Φ, n, ϵ, δ) ≥ e^{−nη−C}P_{μ}(T, \frac{ϕ_{k}}{k} , n, ϵ, δ).

By Theorem 2.1 in [15],

lim_{ϵ→0} lim inf_{n→∞} \frac{1}{n} log P_{μ}(T, \frac{ϕ_{k}}{k} , n, ϵ, δ) = h_{μ}(T) + ∫_{X} \frac{ϕ_{k}}{k} dμ.

Hence by definition, we have

P_{μ}(T, Φ) ≥ lim_{ϵ→0} lim inf_{n→∞} \frac{1}{n} log P_{μ}(T, Φ, n, ϵ, δ) − η ≥ h_{μ}(T) + ∫_{X} \frac{ϕ_{k}}{k} dμ − η.

Note that η is arbitrary. By letting k → ∞ we get

P_{μ}(T, Φ) ≥ lim_{ϵ→0} lim inf_{n→∞} \frac{1}{n} log P_{μ}(T, Φ, n, ϵ, δ) ≥ h_{μ}(T) + Φ_{∗}(μ).  (3.11)

Now we only need to prove the reversed inequality P_{μ}(T, Φ) ≤ h_{μ}(T) + Φ_{∗}(μ).

Take η > 0. For each N, take a set K_{N} as in the proof of Theorem C. Then take N_{0} > 0 such that for any N ≥ N_{0}, μ(K_{N}) > 1 − δ. For any n ≥ N, let F_{n} be a maximal (n, ϵ/2)-separated subset of K_{N}. Then K_{N} ⊆ ∪_{x∈F_{n}}B_{n}(x, ϵ). It means that F_{n} is a (n, ϵ, δ)-spanning set.

By (3.9), if x ∈ K_{N}, then μB_{n}(x, ϵ/2) ≥ exp[−n(h + η)]. Hence, F_{n} contains at most exp[n(h + η)] elements. By (3.10), if x ∈ K_{N}, then ϕ_{n}(x) ≤ n(Φ_{∗}(μ) + η). Now we get

∑_{x∈F_{n}} exp[ϕ_{n}(x)] ≤ ∑_{x∈F_{n}} exp[n(Φ_{∗}(μ) + η)] ≤ exp[n(h + η)] · exp[n(Φ_{∗}(μ) + η)]
= exp[n(h + Φ_{∗}(μ) + 2η)].

Therefore,

P_{μ}(T, Φ, n, ϵ, δ) ≤ exp[n(h + Φ_{∗}(μ) + 2η)].

Consequently,

P_{μ}(T, Φ, δ) ≤ h + Φ_{∗}(μ) + 2η.  (3.12)

Since δ and η are arbitrary, we have

P_{μ}(T, Φ) ≤ h_{μ}(T) + Φ_{∗}(μ),

which is the desired inequality.

Lemma 3.3. Let (X, T) be a TDS and Φ = {ϕ_{n}} a supadditive potential. Fix any positive integer k. For each subset Z we have

P_{Z}(T, Φ) ≥ P_{Z}(T, \frac{1}{k} ϕ_{k}).

Proof. Fix a positive integer k. By Sublemma 3.5 below we have

M(Z, Φ, s, N, ϵ) ≥ e^{−C}M(Z, \frac{1}{k} ϕ_{k}, s + η, N, ϵ).

Therefore

P_{Z}(T, \frac{1}{k} ϕ_{k}) ≤ P_{Z}(T, Φ) + η.

This immediately implies the desired result.

Lemma 3.4. Let (X, T) be a TDS and Φ = {ϕ_{n}} a supadditive potential. Let k be any positive integer and μ ∈ E_{T}. Then for any η > 0, there exist an ϵ_{0} > 0 such that for any 0 < ϵ < ϵ_{0}, n > 0, δ ∈ (0, 1], we have

P_{μ}(T, Φ, n, ϵ, δ) ≥ e^{−nη−C}P_{μ}(T, \frac{ϕ_{k}}{k} , n, ϵ, δ),

and therefore

P_{μ}(T, F) ≥ P_{μ}(T, \frac{f_{k}}{k}).

Remark 3.2. For a subadditive potential F = \{f_{n}\}, the result becomes P_{μ}(T, F) ≤ P_{μ}(T, \frac{f_{k}}{k}).

Proof of Lemma 3.4. By Sublemma 3.5 below, for any small number η > 0, there exist an ϵ_{0} > 0 such that for any 0 < ϵ < ϵ_{0}, n > 0, δ ∈ (0, 1], we have the first inequality of the lemma. Therefore

P_{μ}(T, Φ) ≥ P_{μ}(T, \frac{ϕ_{k}}{k}) − η.

The arbitrariness of η immediately yields the desired result.

Sublemma 3.5. Let (X, T) be a TDS and Φ = {ϕ_{n}} a supadditive potential. Fix any positive integer k. Then for any small number η > 0, there exist an ϵ_{0} > 0 such that for any 0 < ϵ < ϵ_{0} we have

ϕ_{n}(x) ≥ sup_{y∈B_{n}(x,ϵ)} ∑_{i=0}^{n−1} \frac{1}{k} ϕ_{k}(T^{i}y) − nη − C,

where C is a constant independent of η and ϵ.

Proof. Fix a positive integer k. Since \frac{1}{k} ϕ_{k}(x) is a continuous function, for any η > 0, there exist ϵ_{0} > 0 such that for any 0 < ϵ < ϵ_{0},

d(x, y) < ϵ ⇒ d(\frac{1}{k} ϕ_{k}(x), \frac{1}{k} ϕ_{k}(y)) < η.

Using the supadditivity of Φ, as (2.6) we have

ϕ_{n}(x) ≥ ∑_{i=0}^{n−1} \frac{1}{k} ϕ_{k}(T^{i}x) − C,

where C is a constant. Thus

ϕ_{n}(x) ≥ sup_{y∈B_{n}(x,ϵ)} ∑_{i=0}^{n−1} \frac{1}{k} ϕ_{k}(T^{i}y) − nη − C,

which is the desired result.

4 Dimensions of ergodic measure on an average conformal repeller

In this section we give an application of the nonadditive measure-theoretic pressures to average conformal repellers defined in [1]. We use the relations proved in the previous sections among the pressures, entropy, and limits of the nonadditive potentials to prove that the Hausdorff and box dimension of an average conformal repeller is equal to the Hausdorff dimension of an ergodic measure support on it.

Let M be an m-dimensional smooth Riemannian manifold. Let U be an open subset of M and f : U → M be a C^{1} map. Suppose J ⊂ U is a compact f-invariant subset. And let M(f|_{J}) and E(f|_{J}) denote the set of all f−invariant measures and the set of all ergodic measures supported on J respectively.

For x ∈ M and v ∈ T_{x}M, the Lyapunov exponent of v at x is the limit

χ(x, v) = lim_{n→∞} \frac{1}{n} log ∥Df_{x}^{n}(v)∥

if the limit exists. By the Oseledec multiplicative ergodic theorem [21], for μ-almost every point x, every vector v ∈ T_{x}M has a Lyapunov exponent, and they can be denoted by λ_{1}(x) ≤ λ_{2}(x) ≤ ⋯ ≤ λ_{m}(x). Since the Lyapunov exponents are f-invariant, if μ is ergodic, then we can define Lyapunov exponents λ_{1}(μ) ≤ λ_{2}(μ) ≤ ⋯ ≤ λ_{m}(μ), where m = dimM, for the measure.

A compact invariant set J ⊂ M is an average conformal repeller if for any μ ∈ E(f|_{J}), λ_{1}(μ) = λ_{2}(μ) = ⋯ = λ_{m}(μ) > 0. For simplicity we denote by λ(μ) the unique Lyapunov exponent with respect to μ.

Remark 4.1. If a compact f-invariant set J is an average conformal repeller, it is indeed a repeller in the usual way ([9]), that is, f is uniformly expanding on J.

On the other hand, there are average conformal repellers which are not conformal repellers (see an example in [33]).

Remark 4.2. The notion of average conformal repellers is a generalization of the quasi-conformal and asymptotically conformal repellers in [2, 22]. In [1], the authors studied the dimensions of average conformal repellers by using thermodynamic formalism.

Given a set Z ⊂ M, its Hausdorff dimension is defined by

dim_{H}(Z) = inf{s : lim_{ϵ→0} inf_{diamU<ϵ} ∑_{U∈U}(diam U)^{s} = 0},

where U is a cover of Z and diam U = sup{diam U : U ∈ U}. If ν is a probability measure on M, then the Hausdorff dimension of the measure ν is given by

dim_{H}(ν) = inf{dim_{H}(Z) : Z ⊂ M, νZ = 1 }.

The upper and lower box dimensions of Z are defined by

\overline{dim}_{B}(Z) = lim sup_{ϵ→0} \frac{log N(ϵ)}{− log ϵ}  and  \underline{dim}_{B}(Z) = 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 Z. \overline{dim}_{B}(Z) and \underline{dim}_{B}(Z) are also called upper and lower capacities. If \overline{dim}_{B}(Z) = \underline{dim}_{B}(Z), then we simply call this number the box dimension, and denoted it by dim_{B}(Z). Similarly, for any probability measure ν, we have

\overline{dim}_{B}(ν) = lim_{δ→0} inf\{\overline{dim}_{B}(Z) : Z ⊂ M, νZ > 1 − δ\},
\underline{dim}_{B}(ν) = lim_{δ→0} inf\{\underline{dim}_{B}(Z) : Z ⊂ M, νZ > 1 − δ\}.

The upper and lower Ledrappier dimensions ([19, 22]) is given by

\overline{dim}_{L}(ν) = lim_{δ→0} lim sup_{ϵ→0} \frac{log N(ϵ, δ)}{− log ϵ}  and  \underline{dim}_{L}(ν) = 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 M of measure greater than 1 − δ.

It is known that if a compact invariant set J ⊂ M is a conformal repeller for a C^{1} map, then J support an ergodic measure μ such that the Hausdorff or box dimension of measure μ are equal to the Hausdorff or box dimension of the set J, which are known as the variational principle for dimension. And these dimensions can be given by the ratio of the measure-theoretic entropy and the Lyapunov exponent (see e.g. [4, 25, 14]). See also [3, 20] for some recent progresses of variational principle of dimension for non-conformal maps.

Our next theorem is a generalization of the results to average conformal repeller of a C^{1} map f.

Theorem E. Suppose J is an average conformal repeller of a C^{1} map f, then there exists an finvariant ergodic measure μ supported on J such that

D(J) = D(μ) = \frac{h_{μ}(f)}{λ(μ)},

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

In the following, for each f-invariant ergodic measure μ supported on J, we will construct a certain set K whose dimension is equal to the dimension of the measure.

We denote by m(Df(x)) the minimal norm of Df(x), that is, m(Df(x)) = min{∥Df(x)u∥ : u ∈ T_{x}M, ∥u∥ = 1}.

Theorem F. Suppose J is an average conformal repeller of a C^{1} map f, and μ ∈ E(f|_{J}). Let F = {log m(Df^{n}(x))}_{n≥1} and

K = \{x ∈ M : lim_{ϵ→0} lim sup_{n→∞} \frac{− log μ(B_{n}(x, ϵ))}{n} = h_{μ}(T) and lim_{n→∞} \frac{1}{n} log m(Df^{n}(x)) = F_{∗}(μ)\}.

Then we have

D(K) = D(μ),

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

We end the paper by provide a proof of these two theorems.

Proof of Theorem E. In [1], the authors proved that D(J) = s_{0}, where s_{0} is the unique root of the equation P (f, −sF) = 0, where F = {log m(Df^{n}(x))}_{n≥1}. Next, we show that the subadditive topological pressure P (f, −sF) can be attained by an ergodic measure.

By Remark 4.1, we know that the entropy map ν ↦ h_{ν}(f) is upper-semicontinuous on M(f|_{J}). It is also easy to check that ν ↦ −sF_{∗}(ν) is upper-semicontinuous on M(f|_{J}). Hence by compactness of M(f|_{J}), and the variational principle for subadditive topological pressure [8], we have P (f, −sF) = h_{μ_{0}}(f) − sF_{∗}(μ_{0}) for some invariant measure μ_{0} ∈ M(f|_{J}). By ergodic decomposition theorem (see e.g. [27]), we can obtain that P (f, −sF) = h_{μ}(f) − sF_{∗}(μ) for some f-invariant ergodic measure μ.

Note that F_{∗}(μ) = λ(μ) is the unique Lyapunov exponent with respect to μ. Hence D(J) = s_{0} = \frac{h_{μ}(f)}{λ(μ)}. On the other hand, by ([10, Corollary 2]), we can get D(μ) = \frac{h_{μ}(f)}{λ(μ)}. Thus we obtain the equality of the theorem.

Proof of Theorem F. For an f-invariant ergodic measure μ and for each s > 0, by Theorem 4.2 in [1], the set K is equal to the set

\{x ∈ M : lim_{ϵ→0} lim sup_{n→∞} \frac{− log μ(B_{n}(x, ϵ))}{n} = h_{μ}(T) and lim_{n→∞} \frac{−s}{n} ∑_{i=0}^{n−1} ϕ(f^{i}x) = −s ∫ ϕdμ\},

where ϕ(x) = \frac{1}{m} log | det(Df(x))|. Using Theorem B, we have

P_{K}(T, −sϕ) = \overline{CP}_{K}(T, −sϕ) = h_{μ}(f) − s ∫ ϕdμ.  (4.1)

For every subset Z ⊂ J, it is proved in [10] that

dim_{H} Z ≥ t^{∗} and \overline{dim}_{B}Z ≤ s^{∗},

where t^{∗} and s^{∗} are the unique root of the Bowen’s equation P_{Z}(f, −tϕ) = 0 and \overline{CP}_{Z}(f, −sϕ) = 0 respectively. Using this fact and (4.1), we have

dim_{H} K ≥ \frac{h_{μ}(f)}{F_{∗}(μ)} and \overline{dim}_{B}K ≤ \frac{h_{μ}(f)}{F_{∗}(μ)}

since F_{∗}(μ) = ∫ ϕdμ. Note that F_{∗}(μ) is equal to the unique Lyapunov exponent λ(μ) because J is an average conformal repeller. Combining the above inequalities and the result in Theorem E, we have

D(K) = \frac{h_{μ}(f)}{λ(μ)} = D(μ).

Acknowledgements. Authors would like to thank Wen Huang and Pengfei Zhang for giving us the manuscript [16]. They would also like to thank the referee for providing us invaluable suggestions, which led to great improvements of this paper. Cao is partially supported by NSFC (10971151,11125103) and the 973 Project (2007CB814800), Zhao is partially supported by NSFC (11001191), Ph.D. Programs Foundation of Ministry of Education of China (20103201120001) and NSF in Jiangsu province (09KJB110007).

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