The dimensions of non-conformal repeller and average conformal repeller

Jungchao Ban

Department of mathematics, National Hualien University of Education

Hualien 97003, Taiwan, jcban@mail.nhlue.edu.tw

Yongluo Cao*

Department of mathematics, Suzhou University Suzhou 215006, Jiangsu, P.R.China and Institute of Mathematics, Fudan University Shanghai, 200433, P.R. China ylcao@suda.edu.cn, yongluocao@yahoo.com

Huyi Hu

Department of mathematics, Michigan state University East Lansing, MI 48824, USA, hu@math.msu.edu

February 19, 2012

Abstract. In this paper, using thermodynamic formalism for subadditive potential defined in [4], upper bounds of Hausdorff dimension and box dimension of non-conformal repellers are obtained as subadditive Bowen equation. The map f is only needed C^{1}, without additional condition. We also proved that all the upper bounds of Hausdorff dimension in [1, 18, 8] are coincide. This unifies their results. Furthermore we define average conformal repeller and prove that the dimension of average conformal repeller equals to the unique root of sub-additive Bowen equation.

Key words and phrases Hausdorff dimension, Non-conformal repellers, Topological pressure.

02000 Mathematics Subject classification: Primary 37D35; Secondary 37C45.

0* Corresponding author.

1 Introduction.

In the dimension theory of dynamical systems, it is very interesting topic to study the Hausdorff dimension of invariant sets of hyperbolic dynamics. Bowen [3] was the first to express the Hausdorff dimension of an invariant set as a solution of an equation involving topological pressure. Ruelle [13] refined Bowen’s method and obtained the following result. Assume that f is a C^{1+γ} conformal expanding map, Λ is an isolated compact invariant set and f|_{Λ} is topologically mixing, then the Hausdorff dimension of Λ, dim_{H} Λ is given by the unique solution α of the equation

P (f|_{Λ}, −α log ∥D_{x}f∥) = 0  (1.1)

where P (f|_{Λ}, ·) is the topological pressure functional. The smoothness C^{1+γ} was recently relaxed to C^{1} [9].

An estimate from above for the Hausdorff dimension of compact invariant sets for differentiable maps has been given by A.Douady and J.Oesterle [5], and by Ledreppier [11]. For non-conformal dynamical systems there exists only partial results. For example, the Hausdorff dimension of hyperbolic invariant sets was only computed in some special cases. Hu [10] gave an estimate of dimension of non-conformal repeller for C^{2} map. Falconer [6, 7] computed the Hausdorff dimension of a class of non-conformal repellers. Related ideas were applied by Simon and Solomyak [16] to compute the Hausdorff dimension of a class of non-conformal horseshoes in R^{3}.

For C^{1} non-conformal repellers, in [18], Zhang uses singular values of the derivative D_{x}f^{n} for all n ∈ Z^{+}, to define a new equation which involves the limit of a sequence of topological pressure. Then he shows that the unique solution of the equation is an upper bounds of Hausdorff dimension of repeller. In [1], the same problem is considered. Barreira bases on the non-additive thermodynamic formalism which was introduced in [2] and singular value of the derivative D_{x}f^{n} for all n ∈ Z^{+}, and gives an upper bounds of box dimension of repeller under the additional assumptions for which the map is C^{1+γ} and γ-bunched. This automatically implies that for Hausdorff dimension. In [8], Falconer defines topological pressure of sub-additive potential under the condition ∥(D_{x}f)^{−1}∥^{2}∥D_{x}f∥ < 1, which means that f is 1−bunched. They also obtain an upper bounds of Hausdorff dimension of repeller. The questions are whether three bounds as above are the same and whether the upper bounds of box dimension holds true for C^{1} non-conformal repeller?

In this paper, the first, using thermodynamic formalism for sub-additive potential defined in [4], we can obtain upper bounds of Hausdorff dimension and box dimension of non-conformal repellers. The map f is only needed C^{1}, without additional condition. In fact, we prove that the upper bounds of Hausdorff dimension of non-conformal repellers in [18] is the unique root of generalized Bowen equation which relates to subadditive thermodynamic formalism. Furthermore, we proved all the upper bounds in [1, 18, 8] and ours are the same and we can prove that topological pressure in [4] is the same as in [1, 8] in which they need that f is C^{1+γ} and γ-bunched condition. Our result also gives an affirmative answer to problem posed by K.Simon in [15] about an upper bound without assuming the 1-bunched property.

Then we introduce the notion of average conformal repeller. Using thermodynamic formalism for sub-additive potential, we prove that Hausdorff dimension and box dimension of average conformal repellers is the unique root of Bowen equation for sub-additive topological pressure. The map f is only needed C^{1}, without additional condition. Meanwhile, we introduce super-additive potential topological pressure and prove that for special potentials, sub-additive and super-additive topological pressures are same. In [2], Barreira introduces the concept of quasi-conformal repeller by using Markov construction and prove that its dimension is the unique root of the equation obtained by non-additive topological pressure. In [12] introduce the concept of weakly conformal repeller and obtain its dimension using Bowen equation. It is obvious that for C^{1} map quasi-conformal and weakly conformal repeller are average conformal repellers, but reverse is not true. Therefore our result is a generalization of the results in [2, 12].

Next we recall some basic definitions and notations.

Let f : X → X be a continuous map. A set E ⊂ X is called (n, ϵ) separated set with respect to f if x, y ∈ E then d_{n}(x, y) = max_{0≤i≤n−1} d(f^{i}x, f^{i}y) > ϵ. For x ∈ X and r > 0, define

B_{n}(x, r) = {y ∈ X : f^{i}y ∈ B(f^{i}x, r), for all i = 0, ⋯, n − 1}.

If ϕ is a real continuous function on X and n ∈ Z^{+}, let

S_{n}ϕ(x) = ∑_{i=0}^{n−1} ϕ(f^{i}(x)).

We define

P_{n}(ϕ, ϵ) = sup{∑_{x∈E} exp S_{n}ϕ(x) : E is a (n, ϵ) − separated subset of X}.

Then the topological pressure of ϕ is given by

P (f, ϕ) = lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log P_{n}(ϕ, ϵ).

Next we give some properties of P (f, ·) : C(M, R) → R ∪ {∞}.

Proposition 1.1. Let f : M → M be a continuous transformation of a compact metrisable space M. If φ_{1}, φ_{2} ∈ C(X, R), then the followings are true:

(1) P (f, 0) = h_{top}(f).

(2) |P (f, φ_{1}) − P (f, φ_{2})| ≤ ∥φ_{1} − φ_{2}∥.

(3) φ_{1} ≤ φ_{2} implies that P (f, φ_{1}) ≤ P (f, φ_{2}).

Proof. See Walters book [17].

Corollary 1. Let f : M → M be a continuous transformation of a compact metrisable space M. If φ ∈ C(M, R) and φ < 0 then function P (α) = P (f, αφ) is continuous and strictly decreasing in α.

Proof. Let M = max_{x∈M} φ(x) and m = min_{x∈M} φ(x). Then φ ∈ C(M, R) and φ < 0 imply that m ≤ M < 0. If α_{1} < α_{2}, then for all n ∈ ℕ, it has

(α_{2} − α_{1})nm ≤ S_{n}(α_{2}φ)(x) − S_{n}(α_{1}φ)(x) = (α_{2} − α_{1})S_{n}φ(x) ≤ (α_{2} − α_{1})nM.

Thus for ∀ϵ > 0,

e^{(α_{2}−α_{1})nm} × P_{n}(α_{1}φ, ϵ) ≤ P_{n}(α_{2}φ, ϵ) ≤ P_{n}(α_{1}φ, ϵ) × e^{(α_{2}−α_{1})nM}.

It implies that

(α_{2} − α_{1})m + P (f, α_{1}φ) ≤ P (f, α_{2}φ) ≤ P (f, α_{1}φ) + (α_{2} − α_{1})M.

Therefore P (f, αφ) is continuous and strictly monotone decreasing on α.

Another equivalent definition of topological pressure involves open covers.

Definition 1.1. If φ ∈ C(M, R), n ≥ 1 and U is an open cover of M put

p_{n}(f, ϕ, U) = inf{∑_{β} sup_{x∈B} e^{S_{n}ϕ(x)}|β is a finite subcover of ⋁_{i=0}^{n−1} f^{−i}U}.

It is proved [17] that the limit

lim_{n→∞} \frac{1}{n} log p_{n}(f, ϕ, U)

exists and equals to inf_{n>0}\{\frac{1}{n} log p_{n}(f, φ, U)\}.

We have the following Lemma whose proof can be found in [17].

Lemma 1.1. If ϕ ∈ C(M, R), n ≥ 1 and U is an open cover of M, then

lim_{diam(U)→0} lim_{n→∞} \frac{1}{n} log p_{n}(f, ϕ, U) = P (f, ϕ).

A linear map L : ℝ^{n} → ℝ^{n} is said to be expanding if ∥Lv∥ > ∥v∥ for all v ∈ ℝ^{n} and v ≠ 0. Given an expanding linear map L : ℝ^{m} → ℝ^{m}, let λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{m} ≥ 0 be the logarithms of the singular values of L, which are eigenvalues of (L^{∗}L)^{\frac{1}{2}} , counted with their multiplicities, where λ_{m} > 0 because of the expansion. Following [5] we introduce the function

g^{α}(L) = ∑_{i=m−[α]+1}^{m} λ_{i} + (α − [α])λ_{m−[α]},

for any α ∈ [0, m], where [α] is the largest integer ≤ α. g^{α}(L) is continuous and strictly increasing in α. g^{0}(L) = 0 and g^{m}(L) = ∑_{i=1}^{m} λ_{i} = log |Jac(L)|, where Jac(L) is the

Jacobean of L. The map g^{α} has the following super-additive property. If L : ℝ^{m} → ℝ^{m} and L′ : ℝ^{m} → ℝ^{m} are two expanding maps, then

g^{α}(L′L) ≥ g^{α}(L′) + g^{α}(L).  (1.2)

The paper is organized as follows. In Section 2, we develop sub-additive thermodynamics formalism and prove the upper bounds of Hausdorff dimension of non-conformal repellers in [18] is exactly the unique root of the equation of sub-additive topological pressure. In Section 3, we consider the relation between sub-additive thermodynamics formalism defined in [4] and [2, 8], and we obtain for C^{1} non-conformal repeller Λ, upper box dimension is bounded by a value which is the unique solution of the equation of sub-additive topological pressure. This is generalization of the result in [2]. In Section 4, we introduce the definition of average conformal repeller and give related results and the main theorem. In section 5, we develop super-additive thermodynamics formalism and variational principle for super-additive potential. In section 6, we give the proof of main result.

2 A sub-additive thermodynamics formalism

Let f : X → X be a continuous map. A set E ⊂ X is called (n, ϵ) separated set with respect to f if x, y ∈ E then d_{n}(x, y) = max_{0≤i≤n−1} d(f^{i}x, f^{i}y) > ϵ. A sub-additive valuation on X is a sequence of continuous functions ϕ_{n} : M → R such that

ϕ_{m+n}(x) ≤ ϕ_{n}(x) + ϕ_{m}(f^{n}(x)),

we denote it by F = {ϕ_{n}}.

In the following we will define the topological pressure of F = {ϕ_{n}} with respect to f. We define

P_{n}(F, ϵ) = sup{∑_{x∈E} exp ϕ_{n}(x) : E is a (n, ϵ) − separated subset of X}.

Then the topological pressure of F is given by

P (f, F) = lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log P_{n}(F, ϵ).

Let M(X) be the space of all Borel probability measures endowed with the weak* topology. Let M(X, f) denote the subspace of M(X) consisting of all f-invariant measures. For μ ∈ M(X, f), let h_{μ}(f) denote the entropy of f with respect to μ, and let F_{∗}(μ) denote the following limit

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

The existence of the above limit follows from a sub-additive argument. We call F_{∗}(μ) the Lyapunov exponent of F with respect to μ since it describes the exponential growth speed of ϕ_{n} with respect to μ.

In [4], authors proved that the following variational principle

Theorem 2.1. [4] Under the above general setting, we have

P (f, F) = sup{h_{μ}(T) + F_{∗}(μ) : μ ∈ M(X, f)}.

In [2], Barreira used different way to introduce topological pressure for sub-additive potential and proved the variational principle if potential functions satisfies further condition.

Let M be a C^{∞} Riemann manifold, dim M = m. Let U be an open subset of M and let f : U → M be a C^{1} map. Suppose Λ ⊂ U is a compact invariant set on which f is expanding, that is, fΛ = Λ and there is k > 1 such that for all x ∈ Λ and v ∈ T_{x}M,

∥D_{x}fv∥ ≥ k∥v∥,

where ∥.∥ is the norm induced by an adapted Riemannian metric. Let M(f|_{Λ}) denote the all f invariant measures supported on Λ.

If x ∈ Λ, then D_{x}f : T_{x}M → T_{fx}M is a linear map. Denote the logarithms of the singular values of D_{x}f by

λ_{1}(x, f) ≥ λ_{2}(x, f) ≥ ⋯ ≥ λ_{m}(x, f) ≥ log k

and for α ∈ [0, m], write

g^{α}(x, f) = g^{α}(D_{x}f) = ∑_{i=m−[α]+1}^{m} λ_{i}(x, f) + (α − [α])λ_{m−[α]}(x, f).

Since f is C^{1}, the function x ↦ λ_{i}(x, f) and x ↦ g^{α}(x, f) are all continuous.

In fact, fΛ = Λ implies f^{n}Λ = Λ. f^{n} is also expanding on Λ. Let the logarithms of the singular value of D_{x}f^{n} be

λ_{1}(x, f^{n}) ≥ λ_{2}(x, f^{n}) ≥ ⋯ ≥ λ_{m}(x, f^{n}) ≥ n log k

and set

g^{α}(x, f^{n}) = g^{α}(D_{x}f^{n}) = ∑_{i=m−[α]+1}^{m} λ_{i}(x, f^{n}) + (α − [α])λ_{m−[α]}(x, f^{n}).

The functions g^{α}(·, f^{n}) satisfy

g^{α}(x, f^{n+l}) ≥ g^{α}(x, f^{n}) + g^{α}(f^{n}(x), f^{l}).

Define a sequence of functions P_{n} : [0, m] → R as follows:

P_{n}(α) = P (f|_{Λ}, −\frac{1}{n}g^{α}(·, f^{n})).

In [18], author proved that the following result:

Lemma 2.1. [18] For every α ∈ [0, m], the following limit exists

lim_{n→∞} P_{n}(α) = inf_{n∈Z^{+}} P_{n}(α).

Set P^{∗}(α) = lim_{n→∞} P_{n}(α). Then P^{∗} is a continuous and strictly decreasing on [0, m].

Theorem 2.2. [18] Let

D(f, Λ) = max{α ∈ [0, m] : P^{∗}(α) ≥ 0}.

Then

dim_{H} Λ ≤ D(f, Λ).

Remark 1. By variational principle and Ruelle inequality, it has P^{∗}(m) ≤ 0. Since P^{∗}(0) = h(f|_{Λ}) > 0, by Lemma 2.1, it follows that the equation P^{∗}(α) = 0 has an unique solution on [0, m]. By the definition, we have D(f, Λ) is the unique solution of the equation P^{∗}(α) = 0.

In this paper, we first prove the following Proposition.

Proposition 2.1. Suppose {ϕ_{n}(x)} be sub-additive continuous functions sequence on M. Let F = \{ϕ_{n}\}, then we have P (f, F) = lim_{n→∞} P (f, \frac{ϕ_{n}}{n})

Proof. The existence of limit lim_{n→∞} P (f, \frac{ϕ_{n}}{n}) can be found in [18]. First we prove that

P (f, F) ≤ lim_{n→∞} P (f, \frac{ϕ_{n}}{n}).

For a fixed m, let n = ms + l, 0 ≤ l < m. From the subadditivity of {ϕ_{n}}, we have

ϕ_{n}(x) ≤ \frac{1}{m} ∑_{j=0}^{m−1} ∑_{i=0}^{s−2} ϕ_{m}(f^{im+j}(x)) + \frac{1}{m} ∑_{j=0}^{m−1}[ϕ_{j}(x) + ϕ_{m−j+l}(f^{(s−1)m+j}(x))].

Let C_{1} = max_{i=1,··· ,2m−1} max_{x∈X} ϕ_{i}(x). Then it has

ϕ_{n}(x) ≤ ∑_{j=0}^{(sm+l)−1} \frac{1}{m}ϕ_{m}(f^{j}(x)) − \frac{1}{m} ∑_{j=(s−1)m}^{sm−1} ϕ_{m}(f^{j}(x)) + 2C_{1}

≤ ∑_{j=0}^{n−1} \frac{1}{m}ϕ_{m}(f^{j}(x)) + 4C_{1}.

Hence we have

exp(ϕ_{n}(x)) ≤ exp(∑_{j=0}^{n−1} \frac{1}{m}ϕ_{m}(f^{j}(x)) + 4C_{1}).

Thus

P_{n}(F, ϵ) = sup{∑_{x∈E} exp ϕ_{n}(x) : E is a (n, ϵ) − separated subset of X}
≤ P_{n}(\frac{1}{m}ϕ_{m}, ϵ) × exp(4C_{1}).

It implies

P (f, F) ≤ P (f, \frac{1}{m}ϕ_{m}) .

From the arbitrary of m ∈ Z^{+}, we have

P (f, F) ≤ P (f, \frac{1}{m}ϕ_{m}), for all m ∈ Z^{+}.

Therefore

P (f, F) ≤ lim_{n→∞} P (f, \frac{ϕ_{n}}{n}).

Next, we prove that

P (f, F) ≥ lim_{n→∞} P (f, \frac{ϕ_{n}}{n}).

Since f : Λ → Λ is expanding map, h_{μ}(f) is an upper-semi continuous function from M(f|_{Λ}) to R. From variational principle of topological pressure [17], we have that for every k ∈ Z^{+} there exists μ_{2^{k}} ∈ M(f|_{Λ}) such that

P (f|_{Λ}, \frac{1}{2^{k}} ϕ_{2^{k}}) = h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{1}{2^{k}} ϕ_{2^{k}}dμ_{2^{k}}.

Since M(f|_{Λ}) is compact, it implies that μ_{2^{k}} has a subsequence which converges to μ ∈ M(f|_{Λ}). Without loss of generality, suppose that μ_{2^{k}} converges to μ. Using the subadditivity and invariant of μ_{2^{k}}, then we have for every k ∈ ℕ

h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{k}}(x)}{2^{k}}dμ_{2^{k}} ≤ h_{μ_{2^{k}}} (f) + ∫_{Λ} ϕ_{1}(x)dμ_{2^{k}}.

Furthermore for fixed s ∈ ℕ. If k > s, from the subadditivity and invariance of μ_{2^{k}}, it has

h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{k}}(x)}{2^{k}}dμ_{2^{k}} ≤ h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{s}}(x)}{2^{s}}dμ_{2^{k}}.

Since h_{μ}(f) is a upper-semi continuous function, we have

lim_{n→∞} P (f, \frac{ϕ_{n}}{n}) = lim_{k→∞} P (f, \frac{ϕ_{2^{k}}}{2^{k}})
= lim_{k→∞}(h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{k}}(x)}{2^{k}}dμ_{2^{k}})
≤ lim_{k→∞}(h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{s}}(x)}{2^{s}}dμ_{2^{k}})
≤ h_{μ}(f) + ∫_{Λ} \frac{ϕ_{2^{s}}(x)}{2^{s}}dμ.

Since sequence {∫_{Λ} ϕ_{n}(x)dμ} is sub-additive sequence, it has

lim_{n→∞} ∫_{Λ} \frac{ϕ_{n}(x)}{n}dμ = inf_{n≥1}\{∫_{Λ} \frac{ϕ_{n}(x)}{n}dμ\}.

The arbitrariness of s ∈ ℕ implies that

lim_{n→∞} P (f, \frac{ϕ_{n}}{n}) ≤ h_{μ}(f) + lim_{s→∞} ∫_{Λ} \frac{ϕ_{2^{s}}}{2^{s}} (x)dμ.

Hence by variational principle of the sub-additive topological pressure in [4], we have

lim_{n→∞} P (f, \frac{ϕ_{n}}{n}) ≤ h_{μ}(f) + lim_{s→∞} ∫_{Λ} \frac{ϕ_{2^{s}}}{2^{s}} (x)dμ ≤ P (f, F).

This completes the proof of Proposition 2.1.

Theorem 2.3. Let F(α) = {−g^{α}(·, f^{n})}. Then we have P (f, F(α)) = P^{∗}(α).

Proof. For a fixed α, let ϕ_{n}(x) = −g^{α}(x, f^{n}), then it is sub-additive continuous sequence on Λ. By Proposition 2.1 for F(α) = {−g^{α}(x, f^{n})}, we have

P (f, F(α)) = lim_{n→∞} P (f, \frac{ϕ_{n}}{n}) = lim_{n→∞} P (f, −\frac{1}{n}g^{α}(·, f^{n})) = P^{∗}(α).

Theorem 2.4. Let F(α) = {−g^{α}(·, f^{n})}, then we have P (f, F(α)) is continuous and strictly monotone decreasing on α ∈ [0, m]. Thus P (f, F(α)) = 0 has only unique solution in [0, m].

Proof. Let ϕ_{n}(α, x) = −g^{α}(x, f^{n}). If α_{1}, α_{2} ∈ [0, m], α_{1} < α_{2}, then for all n ∈ ℕ, it has

(α_{1} − α_{2})n log k ≥ −ϕ_{n}(α_{2}, x) − (−ϕ_{n}(α_{1}, x)) ≥ (α_{1} − α_{2})n log ∥f∥.

Thus for ∀ϵ > 0,

e^{(α_{1}−α_{2})n log k} × P_{n}(F(α_{1}), ϵ) ≤ P_{n}(F(α_{2}), ϵ) ≤ P_{n}(F(α_{1}), ϵ) × e^{(α_{1}−α_{2})n log ∥f∥}.

It implies that

(α_{1} − α_{2}) log ∥f∥ + P (f, F(α_{1})) ≤ P (f, F(α_{2})) ≤ P (f, F(α_{1})) + (α_{1} − α_{2}) log k.

Therefore P (f, F(α)) is continuous and strictly monotone decreasing on α ∈ [0, m].

One hand, P (f, F(0)) = h_{top}(f) > 0, and on the other hand, by Ruelle inequality [14] and Theorem 2.1, it has P (f, F(m)) ≤ 0. Therefore P (f, F(α)) = 0 has an unique solution in [0, m].

Remark 2. Theorem 2.4 can be deduced from Theorem 2.3 and Lemma 2.1. But for the completeness, we include a different proof.

Corollary 2. D(Λ, f) is the unique solution of equation P (f, F(α)) = 0.

Proof. The proof can be deduced from Theorem 2.3 and Remark 1.

Lemma 2.2. For a fixed n ∈ ℕ, P_{n}(α) = P (f, −\frac{1}{n}g^{α}(·, f^{n})) is a continuous and monotone decreasing function on α ∈ [0, m].

Proof. The proof is analogous to the proof of Theorem 2.4.

By Ruelle-Margulis inequality and variational principle in [17], it has P_{n}(m) = P (f, −\frac{1}{n}g^{m}(·, f^{n})) ≤ 0. Since P_{n}(0) = h(f|_{Λ}) > 0, by Lemma 2.2, it follows that equation P_{n}(α) = 0 has an unique solution. Denote it by α_{n}. Then we have the following proposition.

Theorem 2.5.

inf_{n∈ℕ}α_{n} = D(Λ, f)

Proof. Without loss of generality, we suppose that lim_{n→∞} α_{n} = α^{∗} = inf_{n∈ℕ} α_{n}. Otherwise we can take a subsequence which converges to α^{∗}.

Since

|P (f, −\frac{1}{n}g^{α^{∗}}(·, f^{n})) − P (f, −\frac{1}{n}g^{α_{n}}(·, f^{n}))| ≤ ∥ − \frac{1}{n}g^{α^{∗}}(·, f^{n}) + \frac{1}{n}g^{α_{n}}(·, f^{n})∥
≤ |α^{∗} − α_{n}|∥Df∥,

we have

P (f, F(α^{∗})) = lim_{n→∞} P (f, −\frac{1}{n}g^{α^{∗}}(·, f^{n}))
= lim_{n→∞} P (f, −\frac{1}{n}g^{α_{n}}(·, f^{n})) = 0.

By Corollary 2, we have

D(f, Λ) = α^{∗} = inf_{n∈ℕ}α_{n}.

Now for a fixed n ∈ ℕ, we consider the equation

P̃_{n}(α) = P (f^{n}|_{Λ}, −g^{α}(·, f^{n})) = 0.

It is easy to prove that P̃_{n}(α) is continuous and strictly decreasing on [0, m].

P̃_{n}(0) = h_{top}(f^{n}|_{Λ}) = nh_{top}(f|_{Λ}) ≥ 0

and

P̃_{n}(m) = nP (f|_{Λ}, − log |Jac(D_{x}f)|) ≤ 0.

Hence the equation P̃_{n}(α) = 0 has a unique solution, which we denoted by D_{n}. Applying Lemma 1 in [18] to the expanding map f^{n} yield dim_{H} Λ ≤ D_{n}. So dim_{H} Λ ≤ inf_{n∈Z^{+}} D_{n}. It was proved in [18] that

inf_{n∈Z^{+}} D_{n} ≤ D(f, Λ).

Next we want to prove the reverse inequality, that is say

D(f, Λ) ≤ inf_{n∈Z^{+}} D_{n}.

In order to prove the inequality as above, we firstly prove the following theorem

Proposition 2.2. Suppose {ϕ_{n}(x)} be sub-additive continuous sequence on M. Let

F = \{ϕ_{n}\}, then we have P (f, F) = lim_{k→∞} \frac{1}{k}P (f^{k}, ϕ_{k}).

Proof. For a fixed k ∈ ℕ. It is well known that if E ⊂ M is an (n, ϵ) separated set of f^{k}, then E is an (nk, ϵ) separated set of f. By the definition

P (f^{k}, ϕ_{k}) = lim_{ϵ→∞} lim sup_{n→∞} \frac{1}{n} log sup\{∑_{x∈E} exp(Ŝ_{n}ϕ_{k}(x))
: E is a (n, ϵ) separated set of f^{k}},

where

(Ŝ_{n}ϕ_{k}(x)) = ϕ_{k}(x) + ϕ_{k}(f^{k}x) + ⋯ + ϕ_{k}(f^{(n−1)k}x).

Hence for a fixed m < k, let k = mq + r and C = max_{x∈M} max_{i=1,··· ,2m} ϕ_{i}(x), the subadditivity of ϕ_{n} implies that

ϕ_{k}(x) ≤ \frac{1}{m} ∑_{j=0}^{m−1} ∑_{i=0}^{q−2} ϕ_{m}(f^{im+j}(x)) + \frac{1}{m} ∑_{j=0}^{m−1}[ϕ_{j}(x) + ϕ_{m−j+l}(f^{(q−1)m+j}(x))]

≤ ∑_{i=0}^{k−1} \frac{1}{m}ϕ_{m}(f^{i}(x)) + 4C.

Thus for 1 ≤ j ≤ n − 1, we have

ϕ_{k}(f^{kj}(x)) ≤ ∑_{i=0}^{k−1} \frac{1}{m}ϕ_{m}(f^{i}(f^{kj}(x)) + 4C.

Hence

Ŝ_{n}ϕ_{k}(x) = ϕ_{k}(x) + ϕ_{k}(f^{k}x) + ⋯ + ϕ_{k}(f^{(n−1)k}x)
≤ ∑_{i=0}^{nk−1} \frac{1}{m}ϕ_{m}(f^{i}(x)) + 4nC
= S_{nk}(\frac{1}{m}ϕ_{m})(x) + 4nC.

It gives that

P_{n}(f^{k}, ϕ_{k}, ϵ) ≤ P_{nk}(f, \frac{1}{m}ϕ_{m}, ϵ) × e^{4nC}.

Thus

P (f^{k}, ϕ_{k}) ≤ kP (f, \frac{1}{m}ϕ_{m}) + lim_{n→∞} \frac{1}{n} log e^{4nC}
= kP (f, \frac{1}{m}ϕ_{m}) + 4C.

Therefore

lim_{k→∞} \frac{1}{k}P (f^{k}, ϕ_{k}) ≤ P (f, \frac{1}{m}ϕ_{m}) for all m ∈ Z^{+}.

By Theorem 2.1, it has

lim_{k→∞} \frac{1}{k}P (f^{k}, ϕ_{k}) ≤ lim_{m→∞} P (f, \frac{1}{m}ϕ_{m}) = P (f, F).

Next we prove that

P (f, F) ≤ lim_{k→∞} \frac{1}{k}P (f^{k}, ϕ_{k}).

For a fixed k ∈ ℕ, let n = km + r, 0 ≤ r < k, and let C = max_{x∈M} max_{1≤i≤k} ϕ_{i}(x). For ∀ϵ > 0, by the uniformly continuity of f, there exists δ > 0 such that if E ⊂ M is an (n, ϵ) separated set of f, then E is an (m, δ) separated set of f^{k} and δ → 0 when ϵ → 0. Using the subadditivity of ϕ_{n}, we have

ϕ_{n}(x) ≤ ϕ_{k}(x) + ϕ_{k}(f^{k}(x)) + ⋯ + ϕ_{k}(f^{(m−1)k}(x)) + ϕ_{r}(f^{mk}(x)).

Thus

P_{n}(f, F, ϵ) ≤ P_{m}(f^{k}, ϕ_{k}, δ) × e^{C}.

Hence

P (f, F, ϵ) ≤ \frac{1}{k}P (f^{k}, ϕ_{k}, δ).

It gives that

P (f, F) ≤ \frac{1}{k}P (f^{k}, ϕ_{k}).

From the arbitrary of k, we have that

P (f, F) ≤ lim_{k→∞} \frac{1}{k}P (f^{k}, ϕ_{k}).

Corollary 3. Let F(α) = {−g^{α}(·, f^{k})}, then we have

P (f, F(α)) = lim_{k→∞} \frac{1}{k}P (f^{k}, −g^{α}(·, f^{k})).

Proof. Fixed α, let ϕ_{k}(x) = −g^{α}(x, f^{k}). Using Theorem 2.2 for F(α) = {−g^{α}(x, f^{k})}, we get

P (f, F(α)) = lim_{k→∞} \frac{1}{k}P (f^{k}, −g^{α}(·, f^{k})).

Theorem 2.6.

inf_{n∈ℕ}D_{n} = D(Λ, f)

Proof. Without loss of generality, we suppose that lim_{n→∞} D_{n} = β^{∗} = inf_{n∈ℕ} D_{n}. Otherwise we can take a subsequence which converges to β^{∗}. Since

|\frac{1}{k}P (f^{k}, −g^{β^{∗}}(·, f^{k})) − \frac{1}{k}P (f^{k}, −g^{D_{k}}(·, f^{k}))| ≤ ∥ − \frac{1}{k}g^{β^{∗}}(·, f^{k}) + \frac{1}{k}g^{D_{k}}(·, f^{k})∥
≤ |β^{∗} − D_{k}|∥Df∥,

we have

P (f, F(β^{∗})) = lim_{k→∞} \frac{1}{k}P (f^{k}, −g^{β^{∗}}(·, f^{k}))
= lim_{k→∞} \frac{1}{k}P (f^{k}, −g^{D_{k}}(·, f^{k})) = 0.

Thus

D(f, Λ) = β^{∗} = inf_{k∈ℕ}D_{k}.

In this section, we have proven that for C^{1} non-conformal repeller Λ, D(f, Λ), which is the unique solution of equation P (f, F(α)) = 0, is the upper bounds of Hausdorff dimension of Λ. This is a generalization of the classical result which for C^{1+γ} conformal repeller Λ, dim_{H}Λ is given by the unique solution of the equation P (f|_{Λ}, −α log ∥D_{x}f∥) = 0. Moreover, we prove that

D(f, Λ) = inf_{n∈ℕ}D_{k} = inf_{n∈ℕ}α_{k},

where for each n ∈ ℕ, D_{n} and α_{n} are the unique solutions of equations P̃_{n}(α) = 0 and P_{n}(α) = 0 respectively.

3 Other results of upper bounded estimates of dimension for repeller

Let us first recall Falconer’s definition of topological pressures for sub-additive potentials on mixing repellers. Without loss of generality, we only consider one-sided sub-shift spaces of finite type rather than mixing repellers.

Let (Σ_{A}, σ) be a one-sided sub-shift space over an alphabet {1, …, m}, where m ≥ 2. As usual Σ_{A} is endowed with the metric d(x, y) = m^{−n} where x = (x_{k}), y = (y_{k}) and n is the smallest of the k such that x_{k} ≠ y_{k}. For any admissible string I = i_{1} … i_{n} of length n over the letters {1, …, m}, denote [I] = {(x_{i}) ∈ Σ : x_{j} = i_{j} for 1 ≤ j ≤ n}. The [I] is called an n-th cylinder in Σ_{A}.

Let F be a sub-additive family of continuous potentials define on Σ. Falconer defined the topological pressure of F by

F P (σ, F) = lim_{n→∞} \frac{1}{n} log F P_{n}(σ, F) and F P_{n}(σ, F) = ∑_{[I]} sup_{x∈[I]} e^{ϕ_{n}(x)},

where the summation is taken over the collection of all nth cylinders [I]’s.

It is not so hard to see that in this special case, F P_{n}(σ, F) = P_{n}(σ, F, 1/m), and P_{n}(σ, F, m^{−k}) = F P_{n+k−1}(σ, F) for all k ∈ ℕ. This implies F P (σ, F) is equivalent to our definition P (σ, F).

Now let us turn to Barreira’s approach in defining pressures for sub-additive potentials via open covers.

As in the previous sections, let f be a continuous map acting on a compact metric space (X, d). Let F = {ϕ_{n}}_{n=1}^{∞} be a family of sub-additive continuous functions defined on X. Suppose U is a finite open cover of the space X. For n ≥ 1 we denote by W_{n}(U) the collection of strings U = U_{1} … U_{n} with U_{i} ∈ U. For U ∈ W_{n}(U) we call the integer m(U) = n the length of U and define

X(U) = U_{1} ∩ f^{−1}U_{2} ∩ … ∩ f^{−(n−1)}U_{n}
= {x ∈ X : f^{j−1}x ∈ U_{j} for j = 1, …, n} .

We say that Γ ⊂ ⋃_{n≥1} W_{n}(U) covers X if ⋃_{U∈Γ} X(U) = X. For each U ∈ W_{n}(U), we write e^{ϕ(U)} = sup_{x∈X(U)} e^{ϕ_{n}(x)} when X(U) ≠ ∅ and e^{ϕ(U)} = −∞ otherwise. For s ∈ ℝ, define

M(f, s, F, U) = lim_{n→∞} inf{∑_{U∈Γ} e^{−sm(U)}e^{ϕ(U)}},

where the infimum is taken over all Γ ⊂ ⋃_{j≥n} W_{j}(U) that covers X. Likewise, we define

\underline{M}(f, s, F, U) = lim inf_{n→∞} inf\{∑_{U∈Γ} e^{−sm(U)}e^{ϕ(U)}\},

\overline{M}(f, s, F, U) = lim sup_{n→∞} inf\{∑_{U∈Γ} e^{−sm(U)}e^{ϕ(U)}\},

where the infimum is taken over all Γ ⊂ W_{n}(U) that covers X. Define

P^{⋆}(f, F, U) = inf{s : M(f, s, F, U) = 0} = sup{s : M(f, s, F, U) = +∞},
\underline{CP^{⋆}}(f, F, U) = inf\{s : \underline{M}(f, s, F, U) = 0\} = sup\{s : \underline{M}(f, s, F, U) = +∞\},
\overline{CP^{⋆}}(f, F, U) = inf\{s : \overline{M}(f, s, F, U) = 0\} = sup\{s : \overline{M}(f, s, F, U) = +∞\}.

Define

P^{⋆}(f, F) = lim inf_{diam(U)→0} P^{⋆}(f, F, U),

\underline{CP^{⋆}}(f, F) = lim inf_{diam(U)→0} \underline{CP^{⋆}}(f, F, U),

\overline{CP^{⋆}}(f, F) = lim inf_{diam(U)→0} \overline{CP^{⋆}}(f, F, U).

Barreira named P^{⋆}(f, F) the topological pressure, \underline{CP^{⋆}}(f, F) and \overline{CP^{⋆}}(f, F) the lower and upper topological pressures of F.

Now we consider the connection between P^{⋆}(f, F) and P (f, F). In [4], we prove the following equality.

Proposition 3.1. Assume the topological entropy h(f) < ∞ and the entropy map μ ↦ h_{μ}(f) is upper semi-continuous. Then P^{⋆}(f, F) = P (f, F).

Theorem 3.1. Let M be a C^{∞} Riemann manifold and f : M → M be a C^{1} map. Suppose Λ ⊂ M is a compact invariant set on which f is expanding. Then

P^{⋆}(f, F(α)) = P (f, F(α)).

Proof. Since Λ ⊂ M is a compact invariant set on which f is expanding, it has measure theoretical entropy h_{μ}(f|_{Λ}) is an upper semi-continuous map in M(f|_{Λ}). By proposition 3.1, it has

P^{⋆}(f, F(α)) = P (f, F(α)).

In [1], Barreira proved that if Λ is a repeller of a C^{1+γ} map, for some γ > 0 and f is γ−bunched on Λ, then \overline{dim}_{B} ≤ t^{∗}, where t^{∗} is the unique number of equation P^{⋆}(f, F(α)) = 0. In [1], γ−bunched condition and C^{1+γ} were used to show that it is reasonable to define P^{⋆}(f, F(α)).

Corollary 4. Let M be a C^{∞} Riemann manifold and f : M → M be a C^{1} map. Suppose Λ ⊂ M is a compact invariant set on which f is expanding. Then

dim_{B}Λ ≤ D(Λ, f) and dim_{H}Λ ≤ D(Λ, f),

where D(Λ, f) is the unique solution of equation P (f, F(α)) = 0.

Proof. By Theorem 3.1, we have that if Λ is a repeller of a C^{1} map, then we can define P^{⋆}(f, F(α)) and prove that it is coincidence with P (f, F(α)). It is proved in [1] that \overline{dim}_{B} ≤ t^{∗}, where t^{∗} is the unique solution of equation P^{⋆}(f, F(α)) = 0. Thus we have that t^{∗} = D(Λ, f) which is the unique solution of equation P (f, F(α)) = 0. Therefore we also have the inequality for box dimension.

Remark 3. In [18], Zhang posed a problem whether D(Λ, f) is the upper bound of box dimension of Λ. Corollary as above gives an affirmative answer to the problem. Moreover, our result shows that the subadditive thermodynamic formalism can be apply. In fact we have proven that if Λ is a repeller of a C^{1} map, then the upper bounds of Hausdorff dimension of Λ by Barreira in [1], Falconer in [8] and Zhang in [18] are coincide. This unifies their results and it also shows that bunched condition in [1] and [8] is unnecessary. Our result also gives an affirmative answer to problem posed by K.Simon in [15] about an upper bound without assuming the 1-bunched property.

4 Average conformal repeller

Let M be a C^{∞} Riemann manifold, dim M = d. Let U be an open subset of M and let f : U → M be a C^{1} map. Suppose Λ ⊂ U is a compact expanding invariant set. Let E(f) denote the all ergodic invariant measure supported on Λ respectively. By the Oseledec multiplicative ergodic theorem, for any μ ∈ E(f), we can define Lyapunov exponents λ_{1}(μ) ≤ λ_{2}(μ) ≤ ⋯ ≤ λ_{d}(μ) .

Definition 4.1. An invariant repeller is called average conformal if for any μ ∈ E(f), λ_{1}(μ) = λ_{2}(μ) = ⋯ = λ_{d}(μ) > 0.

It is obvious that a conformal repeller is an average conformal repeller, but reverse isn’t true.

Next we will give main theorem.

Theorem 4.1. (Main Theorem) Let f be C^{1} dynamical system and Λ be an average conformal repeller, then the Hausdorff dimension of Λ is zero t_{0} of t ↦ P (−tF), where

F = {log(m(Df^{n}(x)), x ∈ Λ, n ∈ ℕ}.  (4.3)

where m(A) = ∥A^{−1}∥^{−1}

The proof will be given in section 6.

Theorem 4.2. If Λ be an average conformal repeller, then

lim_{n→∞} \frac{1}{n}(log ∥Df^{n}(x)∥ − log m(Df^{n}(x))) = 0

uniformly on Λ.

Proof. Let

F_{n}(x) = log ∥Df^{n}(x)∥ − log m(Df^{n}(x)), n ∈ ℕ, x ∈ Λ.

It is obviously that the sequence {F_{n}(x)} is a non-negative subadditive function sequence. That is say

F_{n+m}(x) ≤ F_{n}(x) + F_{m}(f^{n}(x)), x ∈ Λ.

Suppose (4.3) is not true, then there exists ϵ_{0} > 0, for any k ∈ ℕ, there exits n_{k} ≥ k and x_{n_{k}} ∈ Λ such that

\frac{1}{n_{k}} F_{n_{k}}(x_{n_{k}}) ≥ ϵ_{0}.

Define measures

μ_{n_{k}} = \frac{1}{n_{k}} ∑_{i=0}^{n_{k}−1} δ_{f^{i}(x_{n_{k}})}.

Compactness of P(f) implies there exists a subsequence of μ_{n_{k}} that converges to measure μ. Without loss of generality, we suppose that μ_{n_{k}} → μ. It is well known that μ is f -invariant. Therefore μ ∈ M(f).

For a fixed m, we have

lim_{k→∞} ∫_{M} \frac{1}{m}F_{m}(x_{n_{k}})dμ_{n_{k}} = ∫_{M} \frac{1}{m}F_{m}(x_{n_{k}})dμ.

It implies

lim_{k→∞} \frac{1}{n_{k}} ∑_{i=0}^{n_{k}−1} \frac{1}{m}F_{m}(f^{i}(x_{n_{k}})) = ∫_{M} \frac{1}{m}F_{m}(x)dμ.

For a fixed m, let n_{k} = ms + l, 0 ≤ l < m. The sub-additivity of {F_{n}} implies that for j = 0, ⋯ m − 1,

F_{n_{k}}(x_{n_{k}}) ≤ F_{j}(x_{n_{k}}) + F_{m}(f^{j}(x_{n_{k}}) + ⋯ + F_{m}(f^{m(s−2)}f^{j}(x_{n_{k}}))
+F_{m−j+l}(f^{m(s−1)}f^{j}(x_{n_{k}}))

Summing j from 0 to m − 1, we get

F_{n_{k}}(x_{n_{k}}) ≤ \frac{1}{m} ∑_{j=0}^{m−1} ∑_{i=0}^{s−2} F_{m}(f^{im+j}(x_{n_{k}}))

+ \frac{1}{m} ∑_{j=0}^{m−1}[F_{j}(x_{n_{k}}) + F_{m−j+l}(f^{(s−1)m+j}(x_{n_{k}}))]

Let C_{1} = max_{i=1,··· ,2m−1} max_{x∈Λ} F_{i}(x).

F_{n_{k}}(x) ≤ ∑_{j=0}^{(sm+l)−1} \frac{1}{m}F_{m}(f^{j}(x)) − \frac{1}{m} ∑_{j=(s−1)m}^{sm−1} F_{m}(f^{j}(x)) + 2C_{1}

≤ ∑_{j=0}^{n_{k}−1} \frac{1}{m}F_{m}(f^{j}(x)) + 4C_{1}.

Hence we have

lim_{k→∞} \frac{1}{n_{k}} F_{n_{k}}(x) ≤ lim_{k→∞} \frac{1}{n_{k}} ∑_{i=0}^{n_{k}−1} \frac{1}{m}F_{m}(f^{i}(x)) = ∫_{M} \frac{1}{m}F_{m}(x)dμ.

The arbitrariness of m ∈ ℕ implies that

lim_{k→∞} \frac{1}{n_{k}} F_{n_{k}}(x) ≤ \frac{1}{m} ∫_{M} F_{m}(x)dμ, ∀m ∈ ℕ.

Hence

lim_{m→∞} \frac{1}{m} ∫_{M} F_{m}(x)dμ ≥ ϵ_{0} > 0.

Then ergodic decomposition theorem [17] implies that there exists μ̃ ∈ E(f) such that

lim_{m→∞} \frac{1}{m} ∫_{M} F_{m}(x)dμ̃ ≥ ϵ_{0} > 0.

On the other hand, from Oseledec theorem and Kingman’s subadditive ergodic theorem, we have lim_{m→∞} \frac{1}{m} ∫_{M} log ∥Df^{m}(x)∥dμ̃ = λ_{d}(μ̃) and lim_{m→∞} \frac{1}{m} ∫_{M} log m(Df^{m}(x))dμ̃ = λ_{1}(μ̃). Therefore

λ_{d}(μ̃) − λ_{1}(μ̃) ≥ ϵ_{0}.

This gives a contradiction to assumption of average conformal.

5 Super-additive variational principle

In this section, we first give the definition of super-additive topological pressure. Then we prove the variational principle for special super-additive potential.

Let f : X → X be a continuous map. A set E ⊂ X is called (n, ϵ) separated set with respect to f if x, y ∈ E then d_{n}(x, y) = max_{0≤i≤n−1} d(f^{i}x, f^{i}y) > ϵ. A super-additive valuation on X is a sequence of functions φ_{n} : M → R such that

φ_{m+n}(x) ≥ φ_{n}(x) + φ_{m}(f^{n}(x)),

we denote it by F = {φ_{n}}.

In the following we will define the topological pressure of F = {φ_{n}} with respect to f. We define

P_{n}^{∗}(F, ϵ) = sup{∑_{x∈E} exp φ_{n}(x) : E is a (n, ϵ) − separated subset of X}.

Then the topological pressure of F is given by

P^{∗}(f, F) = lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log P_{n}(F, ϵ).

For every μ ∈ M(X, f), let F_{∗}(μ) denote the following limit

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

The existence of the above limit follows from a super-additive argument. We call F_{∗}(μ) the Lyapunov exponent of F with respect to μ since it describes the exponential growth speed of φ_{n} with respect to μ.

Theorem 5.1. Let f be C^{1} dynamical system and Λ be an average conformal repeller, and F = {φ_{n}(x)} = {−t log ∥Df^{n}(x)∥} for t ≥ 0 be a super-additive function sequence. Then we have

P^{∗}(f, F) = sup{h_{μ}(T) + F_{∗}(μ) : μ ∈ M(X, f)}.

Proof. First we prove that for any m ∈ ℕ

P^{∗}(f, F) ≥ P (f, \frac{φ_{m}}{m}).

For a fixed m, let n = ms + l, 0 ≤ l < m. From the sup-additivity of {φ_{n}}, we have

φ_{n}(x) ≥ \frac{1}{m} ∑_{j=0}^{m−1} ∑_{i=0}^{s−2} φ_{m}(f^{im+j}(x)) + \frac{1}{m} ∑_{j=0}^{m−1}[φ_{j}(x) + φ_{m−j+l}(f^{(s−1)m+j}(x))].

Let C_{1} = min_{i=1,··· ,2m−1} min_{x∈X} φ_{i}(x). Then it has

φ_{n}(x) ≥ ∑_{j=0}^{(sm+l)−1} \frac{1}{m}φ_{m}(f^{j}(x)) − \frac{1}{m} ∑_{j=(s−1)m}^{sm−1} φ_{m}(f^{j}(x)) + 2C_{1}

≥ ∑_{j=0}^{n−1} \frac{1}{m}φ_{m}(f^{j}(x)) + 4C_{1}.

Hence we have

exp(φ_{n}(x)) ≥ exp(∑_{j=0}^{n−1} \frac{1}{m}φ_{m}(f^{j}(x)) + 4C_{1}).

Thus

P_{n}^{∗}(F, ϵ) = sup{∑_{x∈E} exp φ_{n}(x) : E is a (n, ϵ) − separated subset of X}
≥ P_{n}(\frac{1}{m}φ_{m}, ϵ) × exp(4C_{1}).

It implies

P^{∗}(f, F) ≥ P (f, \frac{1}{m}φ_{m}) .

From the arbitrary of m ∈ Z^{+}, we have

P^{∗}(f, F) ≥ P (f, \frac{1}{m}φ_{m}), for all m ∈ Z^{+}.

By the variational principle in [17], for every μ ∈ M(f), we have

P^{∗}(f, F) ≥ P (f, \frac{1}{m}φ_{m}) ≥ h_{μ}(f) + ∫_{M} \frac{1}{m}φ_{m}(x)dμ, ∀m ∈ ℕ.

Hence we have for every μ ∈ M(f)

P^{∗}(f, F) ≥ h_{μ}(f) + lim_{m→∞} ∫_{M} \frac{1}{m}φ_{m}(x)dμ.

Therefore

P^{∗}(f, F) ≥ sup\{h_{μ}(f) + lim_{m→∞} ∫_{M} \frac{1}{m}φ_{m}(x)dμ, μ ∈ M(f)\}

Let Φ_{n}(x) = −t log m(Df^{n}(x)) for t ≥ 0. Then it is sub-additive. By the theorem in [4], we have

P (f, \{Φ_{n}\}) = sup\{h_{μ}(f) + lim_{m→∞} ∫_{M} \frac{1}{m}Φ_{m}(x)dμ, μ ∈ M(f)\}

By the definitions, −t log m(Df^{n}(x)) ≥ −t log ∥Df^{n}(x)∥ for t ≥ 0 implies that

P^{∗}(f, F) ≤ P (f, {Φ_{n}}).

Theorem 4.3 implies that for any μ ∈ M(f), it has

lim_{m→∞} ∫_{M} \frac{1}{m}Φ_{m}(x)dμ = lim_{m→∞} ∫_{M} \frac{1}{m}φ_{m}(x)dμ.

Therefore

P^{∗}(f, F) = sup\{h_{μ}(f) + lim_{m→∞} ∫_{M} \frac{1}{m}Φ_{m}(x)dμ, μ ∈ M(f)\}.

This completes the proof of theorem.

6 The proof of main theorem

In this section, we will give the proof of main theorem. First we state some known results.

In [1], Barreira prove the following theorem.

Theorem 6.1. If f is a C^{1} expanding map and Λ is a repeller, then

s_{1} ≤ dim_{H}Λ ≤ \underline{dim}_{B}Λ ≤ \overline{dim}_{B}Λ ≤ t_{1}

where s_{1} and t_{1} are the unique roots of the Bowen’s equations P (f, −t log ∥Df(x)∥) = 0 and P (f, −t log m(Df(x))) = 0 respectively.

Since Λ is f-invariant, it is f^{n}-invariant. Hence we have the following corollary.

Corollary 5. If f is a C^{1} expanding map and Λ is a repeller, then

s_{n} ≤ dim_{H}Λ ≤ \underline{dim}_{B}Λ ≤ \overline{dim}_{B}Λ ≤ t_{n}

where s_{n} and t_{n} are the unique roots of the Bowen’s equations P (f^{n}, −t log ∥Df^{n}(x)∥) = 0 and P (f^{n}, −t log m(Df^{n}(x))) = 0 respectively.

Next we prove that the sequences {t_{2^{k}}} and {s_{2^{k}}} are monotone.

Theorem 6.2. The sequence {s_{2^{k}}} is monotone, and

lim_{k→∞} s_{2^{k}} = s_{∗}.

Then we have s_{∗} is the unique root of equation P^{∗}(f, −t{log ∥Df^{n}(x)∥}) = 0.

Proof. First we prove that the sequence {s_{2^{n}}} is monotone increasing. Let φ_{n} = − log ∥(Df^{n}(x)∥ and F = {φ_{n}}. Then it is a sup-additive function sequence. For a fixed k ∈ ℕ,

P_{k}(ϕ, ϵ) = sup{∑_{x∈E} exp S_{n}ϕ(x) : E is a (n, ϵ) − separated subset of X}.

For ∀ϵ > 0, by the uniformly continuity of f, there exists δ > 0 such that if E ⊂ M is an (n, ϵ) separated set of f^{2^{k+1}}, then E is an (2n, δ) separated set of f^{2^{k}} and δ → 0 when ϵ → 0. Using the supadditivity of φ_{n}, the Birkhoff sum S_{n}ϕ_{2^{k+1}} of φ_{2^{k+1}} with respect to f^{2^{k+1}} has the following property:

S_{n}φ_{2^{k+1}}(x) = φ_{2^{k+1}}(x) + φ_{2^{k+1}}(f^{2^{k+1}}x) + ⋯ + φ_{2^{k+1}}(f^{2^{k+1}(n−1)}x)
≥ φ_{2^{k}}(x) + φ_{2^{k}}(f^{2^{k}}x) + φ_{2^{k}}(f^{2^{k+1}}x) + φ_{2^{k}}(f^{2^{k+1}}f^{2^{k}}x)
+ ⋯ + φ_{2^{k}}(f^{2^{k+1}(n−1)}x) + φ_{2^{k}}(f^{2^{k+1}(n−1)}f^{2^{k}}x)
= S_{2n}φ_{2^{k}}(x)

where S_{2n}φ_{2^{k}}(x) is the Birkhoff sum of φ_{2^{k}} with respect to f^{2^{k}}.

Thus

P_{n}(f^{2^{k+1}}, φ_{2^{k+1}}, ϵ) ≥ P_{2n}(f^{2^{k}}, φ_{2^{k}}, δ).

Hence

P (f^{2^{k+1}}, φ_{2^{k+1}}) ≥ 2P (f^{2^{k}}, φ_{2^{k}}).

Therefore if s_{2^{k+1}} is the unique root of Bowen’s equation P (tφ_{2^{k+1}}) = 0, then we have

0 = P (f^{2^{k+1}}, s_{2^{k+1}}φ_{2^{k+1}}) ≥ 2P (f^{2^{k}}, s_{2^{k+1}}φ_{2^{k}}).

Since the function P (f^{2^{k}}, tϕ_{2^{k}}) is monotone decreasing, s_{2^{k}} ≤ s_{2^{k+1}}.

The arbitrariness of k implies that the sequence {s_{2^{k}}} monotone decreasing.

Next we prove that

P^{∗}(f, F) ≥ \frac{1}{k}P (f^{k}, φ_{k}) ∀k ∈ ℕ.

For a fixed k ∈ ℕ, let n = km + r, 0 ≤ r < k, and let C = min_{x∈M} max_{1≤i≤k} ϕ_{i}(x). For ∀ϵ > 0, by the uniformly continuity of f, there exists δ > 0 such that if E ⊂ M is an (n, ϵ) separated set of f, then E is an (m, δ) separated set of f^{k} and δ → 0 when ϵ → 0. Using the sup-additivity of φ_{n}, we have

φ_{n}(x) ≥ φ_{k}(x) + φ_{k}(f^{k}(x)) + ⋯ + φ_{k}(f^{(m−1)k}(x)) + φ_{r}(f^{mk}(x)).

Thus

P_{n}^{∗}(f, F, ϵ) ≥ P_{m}(f^{k}, φ_{k}, δ) × e^{−C}.

Hence

P^{∗}(f, F, ϵ) ≥ \frac{1}{k}P (f^{k}, φ_{k}, δ).

It gives that

P^{∗}(f, F) ≥ \frac{1}{k}P (f^{k}, φ_{k}).

Therefore

P^{∗}(f, F) ≥ \frac{1}{2^{k}} P (f^{2^{k}}, ϕ_{2^{k}}) ∀k ∈ ℕ.

Let tF = {tϕ_{n}(x)}. Then we have

P^{∗}(f, s_{2^{k}}F) ≥ \frac{1}{2^{k}} P (f^{2^{k}}, s_{2^{k}}ϕ_{2^{k}}) = 0 ∀k ∈ ℕ.

The monotone decreasing of P^{∗}(f, tF) with respect to t implies that the unique root s_{∗} of the equation

P^{∗}(f, tF) = 0

satisfies

s_{∗} ≥ s_{2^{k}} ∀k ∈ ℕ.

Thus

s_{∗} ≥ \overline{s} = lim_{k→+∞} s_{2^{k}}.

Next we want to prove that

\overline{s} ≥ s_{∗}.

For a fixed m,

\frac{1}{2^{m}} P (f^{2^{m}}, s_{2^{m}}φ_{2^{m}}) = 0

using the variational principle, for any μ ∈ M(f) ⊂ M(f^{2^{m}}), it has

h_{μ}(f) + \frac{1}{2^{m}} s_{2^{m}} ∫_{M} φ_{2^{m}}dμ = \frac{1}{2^{m}} (h_{μ}(f^{2^{m}}) + s_{2^{m}} ∫_{M} φ_{2^{m}}dμ) ≤ 0.

Let m → ∞, we

h_{μ}(f) + \overline{s} lim_{m→∞} ∫_{M} \frac{1}{2^{m}} φ_{2^{m}}dμ ≤ 0.

Using sup-additive variational principle, we have

P^{∗}(f, \overline{s}\{φ_{n}\}) ≤ 0.

Since P (f, t{φ_{n}}) is strictly monotone decreasing with respect to t, we have

s_{∗} ≤ \overline{s}.

Lemma 6.1. If ϕ_{n}(x) is a subadditive sequence, then

lim_{k→∞} \frac{1}{2^{k}} P (f^{2^{k}}, ϕ_{2^{k}}) ≤ lim_{m→∞} P (f, \frac{ϕ_{2^{m}}}{2^{m}}).

Proof. For a fixed k ∈ ℕ. It is well known that if E ⊂ M is an (n, ϵ) separated set of f^{2^{k}}, then E is an (n2^{k}, ϵ) separated set of f. By the definition

P (f^{2^{k}}, ϕ_{2^{k}}) = lim_{ϵ→∞} lim sup_{n→∞} \frac{1}{n} log sup\{∑_{x∈E} exp(Ŝ_{n}ϕ_{2^{k}}(x))
: E is a (n, ϵ) separated set of f^{2^{k}}},

where

(Ŝ_{n}ϕ_{2^{k}}(x)) = ϕ_{2^{k}}(x) + ϕ_{2^{k}}(f^{2^{k}}x) + ⋯ + ϕ_{2^{k}}(f^{(n−1)2^{k}}x).

Hence for a fixed m < k, let 2^{k} = 2^{m}q + r and C = max_{x∈M} max_{i=1,··· ,2^{m}} ϕ_{i}(x), the subadditivity of ϕ_{n} implies that

ϕ_{2^{k}}(x) ≤ \frac{1}{2^{m}} ∑_{j=0}^{2^{m}−1} ∑_{i=0}^{q−2} ϕ_{2^{m}}(f^{i2^{m}+j}(x)) + \frac{1}{2^{m}} ∑_{j=0}^{2^{m}−1}[ϕ_{j}(x) + ϕ_{2^{m}−j+l}(f^{(q−1)2^{m}+j}(x))]

≤ ∑_{i=0}^{2^{k}−1} \frac{1}{2^{m}} ϕ_{2^{m}}(f^{i}(x)) + 4C.

Thus for 1 ≤ j ≤ n − 1, we have

ϕ_{2^{k}}(f^{2^{k}j}(x)) ≤ ∑_{i=0}^{2^{k}−1} \frac{1}{2^{m}} ϕ_{2^{m}}(f^{i}(f^{2^{k}j}(x)) + 4C.

Hence

Ŝ_{n}ϕ_{2^{k}}(x) = ϕ_{2^{k}}(x) + ϕ_{2^{k}}(f^{2^{k}}x) + ⋯ + ϕ_{2^{k}}(f^{(n−1)2^{k}}x)
≤ ∑_{i=0}^{n2^{k}−1} \frac{1}{2^{m}} ϕ_{2^{m}}(f^{i}(x)) + 4nC
= S_{n2^{k}}(\frac{1}{2^{m}} ϕ_{2^{m}})(x) + 4nC.

It gives that

P_{n}(f^{2^{k}}, ϕ_{2^{k}}, ϵ) ≤ P_{n2^{k}}(f, \frac{1}{2^{m}} ϕ_{2^{m}}, ϵ) × e^{4nC}.

Thus

P (f^{2^{k}}, ϕ_{2^{k}}) ≤ 2^{k}P (f, \frac{1}{2^{m}} ϕ_{2^{m}}) + lim_{n→∞} \frac{1}{n} log e^{4nC}
= 2^{k}P (f, \frac{1}{2^{m}} ϕ_{2^{m}}) + 4C.

Therefore

lim_{k→∞} \frac{1}{2^{k}} P (f^{2^{k}}, ϕ_{2^{k}}) ≤ P (f, \frac{1}{2^{m}} ϕ_{2^{m}}) for all m ∈ Z^{+}.

Hence

lim_{k→∞} \frac{1}{2^{k}} P (f^{2^{k}}, ϕ_{2^{k}}) ≤ lim_{m→∞} P (f, \frac{1}{2^{m}} ϕ_{2^{m}}).

Lemma 6.2.

lim_{n→∞} P (f, \frac{ϕ_{2^{k}}}{2^{k}}) ≤ P (f, F).

Proof. Since f : Λ → Λ is expanding map, h_{μ}(f) is an upper-semi continuous function from M(f|_{Λ}) to R. From variational principle of topological pressure [17], we have that for every k ∈ Z^{+} there exists μ_{2^{k}} ∈ M(f|_{Λ}) such that

P (f|_{Λ}, \frac{1}{2^{k}} ϕ_{2^{k}}) = h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{1}{2^{k}} ϕ_{2^{k}}dμ_{2^{k}}.

Since M(f|_{Λ}) is compact, it implies that μ_{2^{k}} has a subsequence which converges to μ ∈ M(f|_{Λ}). Without loss of generality, suppose that μ_{2^{k}} converges to μ. Using the subadditivity and invariant of μ_{2^{k}}, then we have for every k ∈ ℕ

h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{k}}(x)}{2^{k}}dμ_{2^{k}} ≤ h_{μ_{2^{k}}} (f) + ∫_{Λ} ϕ_{1}(x)dμ_{2^{k}}.

Furthermore for fixed s ∈ ℕ. If k > s, from the subadditivity and invariance of μ_{2^{k}}, it has

h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{k}}(x)}{2^{k}}dμ_{2^{k}} ≤ h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{s}}(x)}{2^{s}}dμ_{2^{k}}.

Since h_{μ}(f) is a upper-semi continuous function, we have

lim_{k→∞} P (f, \frac{ϕ_{2^{k}}}{2^{k}}) = lim_{k→∞}(h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{k}}(x)}{2^{k}}dμ_{2^{k}})
≤ lim_{k→∞}(h_{μ_{2^{k}}} (f) + ∫_{Λ} \frac{ϕ_{2^{s}}(x)}{2^{s}}dμ_{2^{k}})
≤ h_{μ}(f) + ∫_{Λ} \frac{ϕ_{2^{s}}(x)}{2^{s}}dμ.

Since sequence {∫_{Λ} ϕ_{n}(x)dμ} is sub-additive sequence, it has

lim_{n→∞} ∫_{Λ} \frac{ϕ_{n}(x)}{n}dμ = inf_{n≥1}\{∫_{Λ} \frac{ϕ_{n}(x)}{n}dμ\}.

The arbitrariness of s ∈ ℕ implies that

lim_{k→∞} P (f, \frac{ϕ_{2^{k}}}{2^{k}}) ≤ h_{μ}(f) + lim_{s→∞} ∫_{Λ} \frac{ϕ_{2^{s}}}{2^{s}} (x)dμ.

Hence by variational principle of the sub-additive topological pressure in [4], we have

lim_{k→∞} P (f, \frac{ϕ_{2^{k}}}{2^{k}}) ≤ h_{μ}(f) + lim_{s→∞} ∫_{Λ} \frac{ϕ_{2^{s}}}{2^{s}} (x)dμ ≤ P (f, F).

This completes the proof of lemma.

Theorem 6.3. The sequence {t_{2^{n}}} is monotone, and

lim_{n→∞} t_{2^{n}} = t^{∗}

where t^{∗} is the unique root of equation P (f, −t{log m(Df^{n}(x))}) = 0.

Proof. First we prove that the sequence {t_{2^{n}}} is monotone decreasing. Let ϕ_{n} = − log m(Df^{n}(x)). For a fixed k ∈ ℕ,

P_{k}(ϕ, , ϵ) = sup{∑_{x∈E} exp S_{n}ϕ(x) : E is a (n, ϵ) − separated subset of X}.

For ∀ϵ > 0, by the uniformly continuity of f, there exists δ > 0 such that if E ⊂ M is an (n, ϵ) separated set of f^{2^{k+1}}, then E is an (2n, δ) separated set of f^{2^{k}} and δ → 0 when ϵ → 0. Using the subadditivity of ϕ_{n}, the Birkhoff sum S_{n}ϕ_{2^{k+1}} of ϕ_{2^{k+1}} with respect to f^{2^{k+1}} has the following property:

S_{n}ϕ_{2^{k+1}}(x) = ϕ_{2^{k+1}}(x) + ϕ_{2^{k+1}}(f^{2^{k+1}}x) + ⋯ + ϕ_{2^{k+1}}(f^{2^{k+1}(n−1)}x)
≤ ϕ_{2^{k}}(x) + ϕ_{2^{k}}(f^{2^{k}}x) + ϕ_{2^{k}}(f^{2^{k+1}}x) + ϕ_{2^{k}}(f^{2^{k+1}}f^{2^{k}}x)
+ ⋯ + ϕ_{2^{k}}(f^{2^{k+1}(n−1)}x) + ϕ_{2^{k}}(f^{2^{k+1}(n−1)}f^{2^{k}}x)
= S_{2n}ϕ_{2^{k}}(x)

where S_{2n}ϕ_{2^{k}}(x) is the Birkhoff sum of ϕ_{2^{k}} with respect to f^{2^{k}}.

Thus

P_{n}(f^{2^{k+1}}, ϕ_{2^{k+1}}, ϵ) ≤ P_{2n}(f^{2^{k}}, ϕ_{2^{k}}, δ).

Hence

P (f^{2^{k+1}}, ϕ_{2^{k+1}}) ≤ 2P (f^{2^{k}}, ϕ_{2^{k}}).

Therefore if t_{2^{k+1}} is the unique root of Bowen’s equation P (tϕ_{2^{k+1}}) = 0, then we have

0 = P (f^{2^{k+1}}, t_{2^{k+1}}ϕ_{2^{k+1}}) ≤ 2P (f^{2^{k}}, t_{2^{k+1}}ϕ_{2^{k}}).

The monotone decreasing of the function P (f^{2^{k}}, tϕ_{2^{k}}) implies that t_{2^{k}} ≥ t_{2^{k+1}}.

The arbitrariness of k implies that the sequence {t_{2^{k}}} monotone decreasing. Hence limit exists and we denote the limit of this sequence by \overline{t}. From the proof as above, we have

\frac{P (f^{2^{k+1}}, ϕ_{2^{k+1}})}{2^{k+1}} ≤ \frac{P (f^{2^{k}}, ϕ_{2^{k}})}{2^{k}} ≤ ⋯ ≤ \frac{P (f^{2}, ϕ_{2})}{2} ≤ P (f, ϕ).

Next we prove that

P (f, F) ≤ \frac{1}{k}P (f^{k}, ϕ_{k}) ∀k ∈ ℕ.

For a fixed k ∈ ℕ, let n = km + r, 0 ≤ r < k, and let C = max_{x∈M} max_{1≤i≤k} ϕ_{i}(x). For ∀ϵ > 0, by the uniformly continuity of f, there exists δ > 0 such that if E ⊂ M is an (n, ϵ) separated set of f, then E is an (m, δ) separated set of f^{k} and δ → 0 when ϵ → 0. Using the subadditivity of ϕ_{n}, we have

ϕ_{n}(x) ≤ ϕ_{k}(x) + ϕ_{k}(f^{k}(x)) + ⋯ + ϕ_{k}(f^{(m−1)k}(x)) + ϕ_{r}(f^{mk}(x)).

Thus

P_{n}(f, F, ϵ) ≤ P_{m}(f^{k}, ϕ_{k}, δ) × e^{C}.

Hence

P (f, F, ϵ) ≤ \frac{1}{k}P (f^{k}, ϕ_{k}, δ).

It gives that

P (f, F) ≤ \frac{1}{k}P (f^{k}, ϕ_{k}).

Therefore

P (f, F) ≤ \frac{1}{2^{k}} P (f^{2^{k}}, ϕ_{2^{k}}) ∀k ∈ ℕ.  (6.4)

Let tF = {tϕ_{n}(x)}. Then we have

P (f, t_{2^{k}}F) ≤ \frac{1}{2^{k}} P (f^{2^{k}}, t_{2^{k}}ϕ_{2^{k}}) = 0 ∀k ∈ ℕ.

Therefore the unique root t^{∗} of the equation

P (f, tF) = 0

satisfies

t^{∗} ≤ t_{2^{k}} ∀k ∈ ℕ.

Thus

t^{∗} ≤ \overline{t} = lim_{k→+∞} t_{2^{k}}.

Next we want to prove that

\overline{t} ≤ t^{∗}.

From Theorem 6.2 and lemma 6.1, 6.2, we have the sequence \{\frac{1}{2^{k}} P (f^{2^{k}}, ϕ_{2^{k}})\} is monotone decreasing and it converges to P (f, F). By the definition, it is easy to prove that

0 ≤ \frac{P (f^{2^{k}}, \overline{t}ϕ_{2^{k}})}{2^{k}} − \frac{P (f^{2^{k}}, t_{2^{k}}ϕ_{2^{k}})}{2^{k}} ≤ |\overline{t} − t_{2^{k}}|C, ∀k ∈ ℕ,

where C = max_{x∈M} |ϕ_{1}(x)|. Let k → ∞, we have

P (f, \overline{t}F) = 0.

Hence it has, \overline{t} = t^{∗}.

Theorem 6.4. t^{∗} = s_{∗}. Proof. From theorems as above, we have functions

P (f, −t{log m(Df^{n}(x))})

and

P (f, −t{log ∥Df^{n}(x)∥})

coincide and both of them have unique zero points. Therefore

t^{∗} = s_{∗}.

The proof of main theorem:

From Corollary 5 and theorems 6.4 as above, we have

dim_{H}Λ = \underline{dim}_{B}Λ = \overline{dim}_{B} = s_{∗} = t^{∗}.

This completes the proof of main theorem.

Corollary 6. If Λ be an average conformal repeller, then the Hausdorff dimension of Λ is zero t^{∗} of

t ↦ P (−t\frac{1}{d} log(|det(Df)|)),

where d = dimM and t ↦ P (−t\frac{1}{d} log(|det|Df|)) is classical topological pressure. Proof. If Λ be an average conformal repeller, then by Theorem 4.2, we have

lim_{n→∞} \frac{1}{n}(log ∥Df^{n}(x)∥ − log m(Df^{n}(x))) = 0

uniformly on Λ. On the other hand, log(m(Df^{n}(x)) ≤ \frac{1}{d} log(|det(Df^{n}(x))|) ≤ log(|Df^{n}(x)|). Therefore

P (f, −t^{∗}\{log m(Df^{n}(x))\}) = P (f, −t^{∗}\{\frac{1}{d} log |det(Df^{n}(x))|\}) = P (f, −t^{∗}\{log ∥Df^{n}(x)∥\}) = 0.

The additivity of {log ∥Df^{n}(x)∥} implies that

P (f, −t^{∗}\{\frac{1}{d} log |det(Df^{n}(x))|\}) = P (f, −t^{∗} log \frac{1}{d}|det(Df(x)|) = 0.

That is say that t^{∗} is the root of equation P (−t\frac{1}{d} log |det(Df)|) = 0. This gives the proof of corollary.

Acknowledgement. Authors would like to thank Prof.Dejun Feng, Prof.Y.Pesin and Prof.Marcelo Viana for their discussions and suggestions. Authors also would like to thank referees for their suggestions. Ban is partially supported by the National Science Council, R.O.C.(Contract No. NSC 95-2115-M-026-003) and the National Center for Theoretical Sciences, and Cao is partially supported by by NSFC(10571130), NCET, and 973 Project (2007CB814800).

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