A Formula of Packing Pressure of a Factor Map

In this paper, using the notion of packing pressure, we show a formula of packing pressure of a factor map. We also give an application in conformal repellers.


Introduction
Referring to (X, T) as a topological dynamical system (TDS for short) means that X is a compact metric space and T is a continuous self-map.Entropy is an important topological conjugate invariant in dynamical systems.For a factor map π : (X, T) → (Y, S), it is obvious that the entropy of (X, T) is larger than the entropy of (Y, S).In general, the converse is false.For example, (Y, S) is a single point system, and (X, T) have positive topological entropy, then h top (T) > h top (S).A natural question arises whether there exists the converse inequality.In 1971, Bowen [1] considered a factor map π : (X, T) → (Y, S), and showed that h top (T) ≤ h top (S) + sup y∈Y h(T, π −1 y), where h(T, K) denotes the entropy of a compact subset K of X.
In 1973, Ruelle [2] firstly introduced the notion of topological pressure from statistical mechanics to dynamical systems, which is a generalization of entropy.After that, topological pressure becomes one of basic components in thermodynamics and plays an important role in ergodic theory and dynamical systems.
In 1973, Bowen [3] introduced the notion of topological entropy h B top (T, K) for arbitrary subset K in a TDS (X, T) in a way resembling Hausdorff dimension.Later, Pesin and Pitskel [4] extended it to topological pressure of any subset.
In 2012, Fang, Huang, Yi and Zhang [5] proved the following entropy formula for a factor map for any E ⊂ X. Oprocha and Zhang [6] also proved the above formula using open covers.Li , Chen and Zhou [7] extended it to topological pressure.In dimension theory, packing dimension is considered as important as Hausdorff dimension.The set whose Hausdorff dimension is equal to its packing dimension is called Taylor fractal.It is interesting to consider a set is Taylor fractal or not.Recently, Baek, Olsen, Snigireva [8] and Olsen [9] showed that for many sets of divergence points, the packing dimension and the Hausdorff dimension do not coincide.
Inspired by the approach of defining of the topological entropy of non-compact subset, Feng and Huang [10] introduced the notion of packing entropy in dynamical systems, which resembles packing dimension.An understanding of both the topological entropy and the packing entropy of a set provides the basis for substantially better understanding of the underlying dynamical behavior of the set.Similarly, we call the set "dynamical" Taylor fractal if its topological entropy is equal to its packing entropy.In [11], Zhou, Chen and Cheng showed that there are many sets of divergence points which are not "dynamical" Taylor fractals.
Pesin [12] introduced the upper capacity pressure of a set and defined the Caratheodory dimension of a set.The connection between topological pressure and Hausdorff dimension was given by Bowen, who showed that for certain compact sets (quasi-circles) J ⊂ C which arise as invariant sets of fractional linear transformations f of the Riemann sphere, the Hausdorff dimension is the unique root of the equation P J (−tϕ) = 0 where P J is the topological pressure of the map f : J → J, and ϕ is the geometric potential ϕ(z) = log | f (z)|.Barreira and Schmeling [13] have showed that BS dimension is the unique root of topological pressure function.
Inspired by Pesin [12] and Feng and Huang [10], Wang and Chen [14] generalized it to packing topological pressure.In [14], Wang and Chen also introduced packing version of BS dimension and called it BSP dimension.They also showed BSP dimension is the unique root of packing topological pressure function.Recently, Shi [15] obtained Bowen's equation which establishes the relationship of the packing dimension and packing pressure in dynamical systems with some conditions.This illustrates that packing pressure provides a new technique for the study of dimension theory.
In this paper, we will firstly give a formula of the upper capacity pressure for a factor map. Then we show there is a similar relation of packing topological pressure for a factor map.As an application, we obtain that for a factor map with being finite to one or countable to one, the packing dimension is preservable under the factor map.

Preliminaries and Main Result
Let (X, T) be a TDS, and C(X) be the set of all continuous functions on X.For any ϕ ∈ C(X), we set S n ϕ(x) := ∑ n−1 i=0 ϕ(T i (x)).From [16], we can define the following Bowen metric on X, A subset F is said to be an (n, )-spanning set of K, if for any x ∈ K, there exists y ∈ F such that d n (x, y) < .Now, we give a kind of definition of topological pressure resembling Box dimension which is called upper capacity topological pressure.Definition 1. Reference [12] For any ϕ ∈ C(X), and any subset K ⊂ X, The upper capacity topological pressure of K is given by We can also define the upper capacity topological pressure by separated set.The two definitions are equivalent by a standard proof.Now we state the definition as follows without the proof of the equivalence.Definition 2. Reference [12] For any ϕ ∈ C(X), and any subset K ⊂ X, The upper capacity topological pressure of K is defined by Remark that if ϕ = 0, it is called the upper capacity topological entropy of K and written as h(T, K).In the following, we give the definition of packing dimension before the definition of packing topological pressure.Definition 3. If s ≥ 0, and δ > 0 , let where |A| denotes the diameter of A. Since P α δ (Z) decreases when δ decreases, P α 0 (Z) = lim δ→0 P α δ exists.The α-packing measure is defined by The packing dimension of Z is defined by Now, we give the packing topological pressure on arbitrary set, which is similar to packing dimension.
Definition 4. For a set Z ⊂ X and α ∈ R, where P * (Z, N, ) denotes the finite or countable disjoint set family Finally, we define the packing topological pressure P pack (T, ϕ, Z) := lim δ→0 P pack (T, ϕ, Z, δ).The packing entropy of Z is written as h pack (T, Z) as ϕ = 0.
Let (X, T) and (Y, S) be two TDSs with (X, d) and (Y, ρ) be compact metric spaces, respectively.A continuous surjective map π : X → Y satisfying π • T = S • π means that (Y, S) is a factor of (X, T).Now, we state our main result as follows.

Proof of Theorem 1
In this section, we are going to prove the main result of this paper.In the following, for any c > 0 and ϕ ∈ C(Y), we set Var(ϕ, c) = sup x,y∈Y |{ϕ(x) − ϕ(y)| : ρ(x, y) ≤ c}.
To prove the main result, we need the following lemma.The original idea follows from Bowen [1], which compared the entropy of two topological semi-conjugacy systems.Later, Fang, Huang, Yi and Zhang [5] and Oprocha and Zhang [6] extended it to comparing the Bowen entropy of the subset under a factor map.And Li, Chen and Zhou [7] generalized it to Pesin topological pressure.The lemma here is devoted to the study of the upper capacity pressure.Since the proof is standard and similar to the above references, we only present the sketch of the proof.Lemma 1.Let π : (X, T) → (Y, S) be a factor map and ϕ ∈ C(Y).Then for any set E ⊂ X, Proof.(1) Clearly, we have P(S, ϕ, π(E)) ≤ P(T, ϕ • π, E).We only need prove the second inequality.Assume that b := sup y∈Y P(T, ϕ • π, π −1 (y)) < ∞, since b = ∞, the proof is clear.Fix > 0 and τ > 0. For any y ∈ Y choose m(y) ∈ N large enough such that there exists an (m(y), )-spanning set E y of π −1 (y) and Following from the proof of Theorem 2.1(ii) in [7], and choosing δ > 0 small enough, for any y ∈ Y, we can choose t 0 , • • • , t q and ∆ y (t 0 (y)), ∆ y (t 1 (y)) • • • , ∆ y (t q(y) (y)) ∈ Y depending on y recursively such that t 0 (y) = 0 and t s+1 (y) = t s (y) + m(∆ y (t s (y))) until t q+1 (y) ≥ n.For any Then we have For any n ∈ N and sufficiently small δ > 0, we let W n be an (n, δ) spanning set of π(E) with And Combing with Equation (1), we have Moreover, Letting n → ∞, → 0 and τ → 0, we finish the proof.
We introduce the following lemma, which is similar to the packing dimension.
Lemma 2. For any subset Z ⊂ X, we have Proof.Fix any a > P pack (T, ϕ, Z) = lim δ→0 P pack (T, ϕ, Z, δ).There exists some δ > 0 such that So for each i, N large enough, we have M(Z i , a, ϕ, δ, N) < 1. Next, we will prove for any i, P(T, ϕ, Z i ) ≤ a.
For any (N, 3δ) separated set E i of Z i , we can choose disjoint set family {B N (x i , δ)} with x i ∈ E i ⊂ Z i .Furthermore, Let δ → 0, we finish the proof.Now we show the converse inequality.Fix any 0 < t < s < P pack (T, ϕ, Z) and any ∪ ∞ i=1 Z i = Z, we only need to prove that there exists some i ≥ 1 such that P(T, ϕ, Z i ) > t.Since s < P pack (T, ϕ, Z), we can choose δ > 0 such that m * * (Z, s, ϕ, δ) > 0, moreover, there exists α > 0 and some Z i such that m * (Z i , s, ϕ, δ) > α.Furthermore, we can choose N > 0 and a family of disjoint set family {B n j (x j , δ)} with x j ∈ Z i , j ≥ 1, n j ≥ N such that N ϕ(x) ≤ M(Z i , a, ϕ, δ, N) < 1.Then ∑ x∈E i e S N ϕ(x) ≤ e Na .Moreover, P(T, ϕ, Z i , 3δ) ≤ a + Var(ϕ, 3δ).