A note on weighted measure-theoretic pressure

Bin Zhang; Yali Liang; Junjie Zhang

doi:10.1515/math-2024-0125

Article Open Access

A note on weighted measure-theoretic pressure

Bin Zhang , Yali Liang and Junjie Zhang

Published/Copyright: April 30, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this article, we investigate the relations between various types of weighted measure-theoretic pressures and different versions of weighted topological pressures. We show that various types of weighted measure-theoretic pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function.

Keywords: measure-theoretic pressure; weighted pressure; packing topological pressure

MSC 2010: 37A05; 37G05

1 Introductions

Entropies serve as fundamental invariants in characterizing the complexity of dynamical systems. Among their extensions, topological pressure stands out as a non-trivial and natural generalization of topological entropy. The study of these concepts traces back to Kolmogorov, who introduced measure-theoretic entropy as an isomorphic invariant for measure-preserving dynamical systems [1,2]. Shortly afterward, Adler et al. defined topological entropy via open covers as a conjugate invariant [3], while Bowen [4] and Dinaburg [5] independently provided equivalent formulations using separated and spanning sets.

Building on ideas from statistical mechanics, Ruelle [6] introduced topological pressure for continuous functions under Z + -actions on compact spaces, establishing a variational principle under expansivity and the specification property. Walters [7] later generalized this result to conditions without such constraints. Further developments by Bowen extended topological entropy to arbitrary sets in topological dynamical systems using a Hausdorff dimension-like approach [8]. Pesin and Pitskel’ [9] subsequently generalized this to noncompact sets, proving a variational principle under additional conditions. These concepts – topological pressure, variational principles, and equilibrium states – play a pivotal role in statistical mechanics, ergodic theory, and dynamical systems [10].

As key components of thermodynamic formalism [11], topological pressure and its associated variational principle and equilibrium measures are indispensable in the dimension theory of dynamical systems. They provide essential tools for analyzing the dimension of invariant sets and measures in conformal dynamics [10,12,13]. Recent work by Feng and Huang [14] introduced weighted topological pressure with a corresponding variational principle, while Tsukamoto [15] proposed alternative definitions of weighted topological entropy and pressure. The equivalence between these frameworks remains non-trivial and can be viewed as a topological generalization of the Bedford-McMullen carpet dimension formula.

In this article, inspired by Feng and Huang [14], we introduce a generalized measure-theoretic pressure for factor maps between topological dynamical systems, extending the work of Pesin and Pitskel [9]. Adopting their approach, we define a weighted measure-theoretic pressure analogous to Hausdorff and packing measures, aiming to establish connections between Pesin-Pitskel pressure, packing-weighted pressure, and measure-theoretic entropy.

2 Preliminaries

Let k ≥ 2 . Assume that ( X i , d i ) , i = 1 , … , k , are compact metric spaces, and ( X i , T i ) are topological dynamical systems. Moreover, assume that for each 1 ≤ i ≤ k − 1 , ( X i + 1 , T i + 1 ) is a factor of ( X i , T i ) with a factor map π i : X i → X i + 1 ; in other words, π 1 , … , π k − 1 are continuous maps such that the following diagrams commute.

$2$

For convenience, we use π 0 as the identity map on X 1 . Define τ i : X 1 → X i + 1 by τ i = π i ∘ π i − 1 ∘ … ∘ π 0 , for i = 0 , 1 , … , k − 1 . Let M ( X i , T i ) denote the set of all T i -invariant Borel probability measures on X i and E ( X i , T i ) denote the set of ergodic measures. Fix a = ( a 1 , a 2 , … , a k ) ∈ R k with a 1 > 0 and a i ≥ 0 for i ≥ 2 . For μ ∈ M ( X 1 , T 1 ) , we call

h μ a ( T 1 ) ≔ ∑ i = 1 k a i h μ ∘ τ i − 1 − 1 ( T i )

the a -weighted measure-theoretic entropy of μ with respect to T 1 , or simply, the a -weighted entropy of μ , where h μ ∘ τ i − 1 − 1 ( T i ) denotes the measure-theoretic entropy of μ ∘ τ i − 1 − 1 with respect to T i . Let C ( X 1 , R ) denote the set of all continuous functions of X 1 , and let P μ a , B ( f ) and P μ a , K B ( f ) denote, respectively, the Pesin-Pitskel pressure of μ (see Definition 2.4) and the Pesin-Pitskel pressure of μ in the sense of Katok (see Definition 2.5). Thus, we try to find relationships between these notions of different weighted pressure.

Definition 2.1

[14] (a-Weighted Bowen ball) For x ∈ X 1 , n ∈ N , ε > 0 , denote

B n a ( x , ε ) = { y ∈ X 1 : d i ( T i j τ i − 1 x , T i j τ i − 1 y ) < ε for 0 ≤ j ≤ ⌈ ( a 1 + … + a i ) n ⌉ − 1 , i = 1 , … , k } ,

where ⌈ u ⌉ denotes the least integer ≥ u . We call B n a ( x , ε ) the n -th a-weighted Bowen ball of radius ε centered at x .

2.1 Weighted topological pressure

Definition 2.2

Let Z ⊂ X 1 be a nonempty set. Given n ∈ N , α ∈ R , ε > 0 , and f ∈ C ( X 1 , R ) , define

M a ( N , α , ε , Z , f ) = inf ∑ i exp − α n i + 1 a 1 S ⌈ a 1 n i ⌉ f ( x i ) : Z ⊂ ⋃ i B n i a ( x i , ε ) ,

where the infimum is taken over all finite or countable collections of { B n i a ( x i , ε ) } i such that x i ∈ X , n i ≥ N , and ⋃ i B n i a ( x i , ε ) ⊃ Z . Likewise, we define

R a ( n , α , ε , Z , f ) = inf ∑ i exp − α n + 1 a 1 S ⌈ a 1 n ⌉ f ( x i ) : Z ⊂ ⋃ i B n a ( x i , ε ) ,

where the infimum is taken over all finite or countable collections of { B n a ( x i , ε ) } i such that x i ∈ X , n ∈ N , and ⋃ i B n a ( x i , ε ) ⊃ Z . Define

M a , P ( N , α , ε , Z , f ) = sup ∑ i exp − α n + 1 a 1 S ⌈ a 1 n i ⌉ f ( x i ) ,

where the supremum is taken over all finite or countable pairwise disjoint families { B ¯ n i a ( x i , ε ) } such that x i ∈ Z , n i ≥ N for all i , where

B ¯ n i a ( x i , ε ) = { y ∈ X 1 : d n i ( T i j τ i − 1 x , T i j τ i − 1 y ) ≤ ε for 0 ≤ j ≤ ⌈ ( a 1 + … + a i ) n ⌉ − 1 , i = 1 , … , k } .

Let

M a ( α , ε , Z , f ) = lim N → ∞ M a ( N , α , ε , Z , f ) , R ̲ a ( α , ε , Z , f ) = liminf N → ∞ R a ( N , α , ε , Z , f ) , R ¯ a ( α , ε , Z , f ) = limsup N → ∞ R a ( N , α , ε , Z , f ) , M a , P ( α , ε , Z , f ) = lim N → ∞ M a , P ( N , α , ε , Z , f ) .

Define

M a , P ( α , ε , Z , f ) = inf ∑ i = 1 ∞ M a , P ( α , ε , Z i , f ) : Z ⊂ ⋃ i = 1 ∞ Z i .

It is routine to check that when α goes from −∞ to +∞, the quantities

M a ( α , ε , Z , f ) , M ̲ a ( α , ε , Z , f ) , M ¯ a ( α , ε , Z , f ) , M a , P ( α , ε , Z , f )

jump from + ∞ to 0 at unique critical values, respectively. Hence, we can define the numbers

P a , B ( ε , Z , f ) = sup { α : M a ( α , ε , Z , f ) = + ∞ } = inf { α : M a ( α , ε , Z , f ) = 0 } , C P ̲ a ( ε , Z , f ) = sup { α : R ̲ a ( α , ε , Z , f ) = + ∞ } = inf { α : R ̲ a ( α , ε , Z , f ) = 0 } , C P ¯ a ( ε , Z , f ) = sup { α : R ¯ a ( α , ε , Z , f ) = + ∞ } = inf { α : R ¯ a ( α , ε , Z , f ) = 0 } , P a , P ( ε , Z , f ) = sup { α : M a , P ( α , ε , Z , f ) = + ∞ } = inf { α : M a , P ( α , ε , Z , f ) = 0 } .

Definition 2.3

We call the following quantities:

P a , B ( Z , f ) = lim ε → 0 P a , B ( ε , Z , f ) , C P ̲ a ( Z , f ) = lim ε → 0 C P ̲ a ( ε , Z , f ) , C P ¯ a ( Z , f ) = lim ε → 0 C P ¯ a ( ε , Z , f ) , P a , P ( Z , f ) = lim ε → 0 P a , P ( ε , Z , f ) ,

weighted Pesin-Pitskel, weighted lower capacity, weighted upper capacity, and weighted packing topological pressures of T 1 on the set Z with respect to f , respectively.

2.2 Weighted measure-theoretic pressure

Definition 2.4

We call the following quantities:

P μ a , B ( f ) ≔ lim ε → 0 lim δ → 0 inf { P a , B ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } = lim ε → 0 lim δ → 0 inf { P a , B ′ ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } , C P ̲ μ a ( f ) ≔ lim ε → 0 lim δ → 0 inf { C P ̲ a ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } = lim ε → 0 lim δ → 0 inf { C P ̲ a , ′ ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } , C P ¯ μ a ( f ) ≔ lim ε → 0 lim δ → 0 inf { C P ¯ a ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } = lim ε → 0 lim δ → 0 inf { C P ¯ a , ′ ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } , P μ a , P ( f ) ≔ lim ε → 0 lim δ → 0 inf { P a , P ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } = lim ε → 0 lim δ → 0 inf { P a , P ′ ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } .

Definition 2.5

Let Z ⊂ X be a nonempty set. Given μ ∈ ℳ ( X ) , n ∈ N , α ∈ R , ε > 0 , 0 < δ < 1 , and f ∈ C ( X 1 , R ) , define

M μ a ( N , α , ε , δ , f ) = inf ∑ i exp − α n i + 1 a 1 S ⌈ a 1 n i ⌉ f ( x i ) : μ ( ⋃ i B n i a ( x i , ε ) ) ≥ 1 − δ ,

where the infimum is taken over all finite or countable collections of { B n i a ( x i , ε ) } i such that x i ∈ X , n i ≥ N , and μ ( ⋃ i B n i a ( x i , ε ) ) ≥ 1 − δ . Likewise, we define

R μ a ( n , α , ε , δ , f ) = inf ∑ i exp − α n + 1 a 1 S ⌈ a 1 n ⌉ f ( x i ) : μ ( ⋃ i B n ( x i , ε ) ) ≥ 1 − δ ,

where the infimum is taken over all finite or countable collections of { B n a ( x i , ε ) } i such that x i ∈ X , n ∈ N and μ ( ⋃ i B n a ( x i , ε ) ) ≥ 1 − δ . Let

M μ a ( α , ε , δ , f ) = lim N → ∞ M μ a ( N , α , ε , δ , f ) , M ̲ μ a ( α , ε , δ , f ) = liminf N → ∞ R μ a ( N , α , ε , δ , f ) , M ¯ μ a ( α , ε , δ , T , f ) = limsup N → ∞ R μ a ( N , α , ε , δ , f ) .

Define

M μ a , P ( α , ε , δ , f ) = inf ∑ i = 1 ∞ M a , P ( α , ε , Z i , f ) : μ ⋃ i = 1 ∞ Z i ≥ 1 − δ .

Thus, when α goes from −∞ to +∞, the quantities

M μ a ( α , ε , δ , f ) , M ̲ μ a ( α , ε , δ , f ) , M ¯ μ a ( α , ε , δ , f ) , and M μ a , P ( α , ε , δ , f )

jump from +∞ to 0 at unique critical values, respectively. Hence, we can define the numbers

P μ a , K B ( ε , δ , f ) = sup { α : M μ a ( α , ε , δ , f ) = + ∞ } = inf { α : M μ a ( α , ε , δ , f ) = 0 } , C P ̲ μ a , K ( ε , δ , f ) = sup { α : R ̲ μ a ( α , ε , δ , f ) = + ∞ } = inf { α : R ̲ μ a ( α , ε , δ , f ) = 0 } , C P ¯ μ a , K ( ε , δ , f ) = sup { α : R ¯ μ a ( α , ε , δ , f ) = + ∞ } = inf { α : R ¯ μ a ( α , ε , δ , f ) = 0 } , P μ a , K P ( ε , δ , f ) = sup { α : M μ a , P ( α , ε , δ , f ) = + ∞ } = inf { α : M μ a , P ( α , ε , δ , f ) = 0 } .

Definition 2.6

We call the following quantities:

P μ a , K B ( f ) = lim ε → 0 lim δ → 0 P μ a , K B ( ε , δ , f ) , C P ̲ μ a , K ( f ) = lim ε → 0 lim δ → 0 C P ̲ μ a , K ( ε , δ , f ) , C P ¯ μ a , K ( f ) = lim ε → 0 lim δ → 0 C P ¯ μ a , K ( ε , δ , f ) , P μ a , K P ( f ) = lim ε → 0 lim δ → 0 P μ a , K P ( ε , δ , f ) ,

weighted Pesin-Pitskel, weighted lower capacity, weighted upper capacity, and weighted packing pressures of μ in the sense of Katok with respect to f , respectively.

Let f ∈ C ( X 1 , R ) and μ ∈ ℳ ( X ) . The measure-theoretic lower and upper local pressures of x ∈ X 1 with respect to μ and f are defined by

P ̲ μ a ( x , f ) ≔ lim ε → 0 liminf n → ∞ − log μ ( B n a ( x , ε ) ) + 1 a 1 S ⌈ a 1 n ⌉ f ( x ) n , P ¯ μ a ( x , f ) ≔ lim ε → 0 limsup n → ∞ − log μ ( B n a ( x , ε ) ) + 1 a 1 S ⌈ a 1 n ⌉ f ( x ) n .

Definition 2.7

The measure-theoretic lower and upper local pressures of μ with respect to f are defined as

P ̲ μ a ( f ) ≔ ∫ P ̲ μ a ( x , f ) d μ , P ¯ μ a ( f ) ≔ ∫ P ¯ μ a ( x , f ) d μ .

Now, we state our main results as follows.

Theorem 2.1

Let f ∈ C ( X 1 , R ) and μ be a non-atomic Borel ergodic measure on X 1 . Then,

P μ a , K B ( f ) = C P ̲ μ a , K ( f ) = C P ¯ μ a , K ( f ) = P μ a , K P ( f ) = P μ a , B ( f ) = C P ̲ μ a ( f ) = C P ¯ μ a ( f ) = P μ a , P ( f ) = h μ a ( T 1 ) + ∫ X 1 f d μ .

3 Proof of Theorem 2.1

To prove the main results, we first give a weighted topological pressure inequality as follows.

Proposition 3.1

For any f ∈ C ( X 1 , R ) and any subset Z ⊂ X 1 ,

P a , B ( Z , f ) ≤ P a , P ( Z , f ) ≤ C P ¯ a ( Z , f ) .

Proof

We first show that P a , B ( Z , f ) ≤ P a , P ( Z , f ) . Suppose that P a , B ( Z , f ) > s > − ∞ . For any ε > 0 and n ∈ N , let

ℱ n , ε a = { ℱ a : ℱ a = { B ¯ n a ( x i , ε ) } , x i ∈ Z , and ℱ a is a disjoint family } .

Take ℱ a ( N , ε , Z ) ∈ ℱ N , ε a such that ∣ ℱ a ( N , ε , Z ) ∣ = max ℱ a ∈ ℱ N , ε a { ∣ ℱ a ∣ } , where ∣ ℱ a ∣ denotes the cardinality of ℱ . We denote ℱ a ( N , ε , Z ) = { B ¯ N a ( x i , ε ) , i = 1 , … , ∣ ℱ a ( N , ε , Z ) ∣ } . It is easy to check that

Z ⊂ ⋃ i = 1 ∣ ℱ a ( N , ε , Z ) ∣ B N a ( x i , 2 ε + δ ) , ∀ δ > 0 .

Then, for any s ∈ R ,

M a ( N , s , 2 ε + δ , Z , f ) ≤ e − s N ∑ i = 1 ∣ ℱ ( N , ε , Z ) ∣ exp 1 a 1 S ⌈ a 1 N ⌉ f ( x i ) ≤ M a , P ( N , s , ε , Z , f ) .

It thus follows that M a ( s , 2 ε + δ , Z , f ) ≤ M a , P ( s , ε , Z , f ) . By Definition 3.1, we can obtain that M a , P ( α , ε , Z , f ) ≤ M a , P ( s , ε , Z , f ) ; thus, we have M a ( s , 2 ε + δ , Z , f ) ≤ M a , P ( s , ε , Z , f ) . Since P a , B ( Z , f ) > s > − ∞ , M a ( s , 2 ε + δ , Z , f ) ≥ 1 when ε and δ are small enough. Thus, M a , P ( s , ε , Z , f ) ≥ 1 . This implies that P a , P ( ε , Z , f ) ≥ s for ε small enough. Hence, P a , P ( Z , f ) ≥ s and P a , B ( Z , f ) ≤ P a , P ( Z , f ) .

Next, we shall show P a , P ( Z , f ) ≤ C P ¯ a ( Z , f ) .

Without generality, we assume P a , P ( Z , f ) > − ∞ . Choose − ∞ < t < s < P a , P ( Z , f ) . Then, there exists δ > 0 , such that for any ε ∈ ( 0 , δ ) , P a , P ( ε , Z , f ) > s and M a , P ( s , ε , Z , f ) ≥ M a , P ( s , ε , Z , f ) = ∞ . Hence, for any N ∈ N , there exists a countable pairwise disjoint family { B ¯ n i a ( x i , ε ) } such that x i ∈ Z , n i ≥ N for all i , and ∑ i exp − n i s + 1 a 1 S ⌈ a 1 n i ⌉ f n i ( x ) > 1 . For each k , let

m k = { x i : n i = k } .

Then,

∑ k = N ∞ ∑ x ∈ m k exp 1 a 1 S ⌈ a 1 n k ⌉ f ( x ) e − k s > 1 .

It is easy to check that there exists k ≥ N such that

∑ x ∈ m k exp 1 a 1 S ⌈ a 1 n k ⌉ f ( x ) e − k t ≥ 1 − e t − s

(otherwise, ∑ k = N ∞ ∑ x ∈ m k exp 1 a 1 S ⌈ a 1 n k ⌉ f ( x ) e − k s ≤ 1 ). Fixing a collection B k a y i , ε 2 i ∈ I such that Z ⊂ ⋃ i ∈ I B k a y i , ε 2 , where I is at most countable, it is not difficult to check that for any x 1 , x 2 ∈ m k there exists different y 1 and y 2 such that x i ∈ B a y i , ε 2 , i = 1, 2. Then,

R a k , t , ε 2 , Z , f ≥ ∑ x ∈ m k exp 1 a 1 S ⌈ a 1 n i ⌉ f k ( x ) e − k t ≥ 1 − e t − s .

Hence,

R ¯ a t , ε 2 , Z , f = limsup k → ∞ R a k , t , ε 2 , Z , f ≥ 1 − e t − s > 0 .

Thus, C P ¯ a ε 2 , Z , f ≥ t . Letting ε → 0 yields C P ¯ a ( Z , f ) ≥ t . Since t ∈ ( − ∞ , P a , P ( Z , f ) ) , it follows that P a , P ( Z , f ) ≤ C P ¯ a ( Z , f ) .□

Proposition 3.2

Let μ ∈ ℳ ( X 1 ) and f ∈ C ( X 1 , R ) . Then,

P μ a , K B ( f ) = P μ a , B ( f ) , C P ̲ μ a , K ( f ) = C P ̲ μ a ( f ) , C P ¯ μ a , K ( f ) ≤ C P ¯ μ a ( f ) , P μ a , K P ( f ) = P μ a , P ( f ) .

Proof

We shall show that P μ a , K B ( f ) ≤ P μ a , B ( f ) . For any N ∈ N , α ∈ R , ε > 0 , 0 < δ < 1 , and Z with μ ( Z ) ≥ 1 − δ ,

M μ a ( N , α , ε , δ , f ) ≤ M a ( N , α , ε , Z , f ) .

Letting N → ∞ yields

M μ a ( α , ε , δ , f ) ≤ M a ( α , ε , Z , f ) .

This shows that

P μ a , K B ( ε , δ , f ) ≤ P a , B ( ε , Z , f ) ,

and consequently,

P μ a , K B ( ε , δ , f ) ≤ inf { P a , B ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } .

Letting δ → 0 and ε → 0 , the desired inequality follows. We can prove similarly C P ̲ μ a , K ( f ) ≤ C P ̲ μ a ( f ) and C P ¯ μ a , K ( f ) ≤ C P ¯ μ a ( f ) .

To prove P μ a , K B ( f ) ≥ P μ a , B ( f ) , let a = P μ a , K B ( f ) . For any s > 0 , there exists ε ′ > 0 such that

lim δ → 0 P μ a , K B ( ε , δ , f ) < a + s , ∀ ε < ε ′ .

It follows that for any ε ∈ ( 0 , ε ′ ) , there exists δ ε so that

P μ a , K B ( ε , δ , f ) < a + s , ∀ δ < δ ε .

This implies that lim n → ∞ M μ a ( n , a + s , ε , δ , f ) = 0 . For any N ∈ N , we can find a sequence of δ N , m with lim m → 0 δ N , m = 0 and a collection of { B n i a ( x i , ε ) } i ∈ I N , m such that x i ∈ X , n i ≥ N , μ ( ⋃ i ∈ I N , m B n i a ( x i , ε ) ) ≥ 1 − δ N , m , and

∑ i ∈ I N , m exp − ( a + s ) n i + 1 a 1 S ⌈ a 1 n i ⌉ f ( x ) ≤ 1 2 m .

Let

Z N = ⋃ m ∈ N ⋃ i ∈ I N , m B n i a ( x i , ε ) .

Then, μ ( Z N ) = 1 and

M a ( N , a + s , ε , Z N , f ) ≤ 1 .

Let Z ε = ⋂ N ∈ N Z N . Thus, μ ( Z ε ) = 1 and

M a ( N , a + s , ε , Z ε , f ) ≤ M a ( N , a + s , ε , Z N , f ) ≤ 1 , ∀ N ∈ N .

It follows that

P a , B ( ε , Z ε , f ) ≤ a + s .

Therefore,

P μ a , B ( f ) = lim ε → 0 liminf δ → 0 { P a , B ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } ≤ a + s .

The arbitrariness of s then implies the desired inequality. To prove C P ̲ μ a , K ( f ) ≥ C P ̲ a ( f ) , let a = C P ̲ μ a , K ( f ) . For any s > 0 , there exists ε ′ > 0 such that for any ε ∈ ( 0 , ε ′ ) , there exists δ ε so that

liminf N → ∞ R μ a ( N , a + s , ε , δ , f ) = 0 , ∀ δ < δ ε .

Fix δ ∈ ( 0 , δ ε ) . For any m ∈ N , we have

liminf N → ∞ R μ a N , a + s , ε , δ 2 m , f = 0 .

Then, for every m ∈ N , there exists a family { B k m a ( x i , ε ) } i ∈ I m with μ ( ⋃ i ∈ I m B k m a ( x i , ε ) ) ≥ 1 − δ 2 m such that

∑ i ∈ I m e − ( a + s ) k m + 1 a 1 S ⌈ a 1 k m ⌉ f ( x i ) ≤ 1 .

Let Z δ = ⋂ m ∈ N ⋃ i ∈ I m B k m a ( x i , ε ) . Then, μ ( Z δ ) ≥ 1 − δ . It is easy to check that

liminf N → ∞ R a ( N , a + s , ε , Z δ , f ) ≤ 1 .

Thus,

C P ̲ a ( ε , Z δ , f ) ≤ a + s .

This implies that C P ̲ μ a ( f ) ≤ a + s , and the desired inequality follows from the arbitrariness of s .

Similarly, we can obtain

C P ¯ a ( f ) ≤ C P ¯ a , K ( f ) .

We now show the fourth equality. We first prove that P μ a , P ( f ) ≥ P μ a , K P ( f ) . For any s < P μ a , K P ( f ) , there exists ε ′ , δ ′ > 0 such that

P μ a , K P ( ε , δ , f ) > s , ∀ ε ∈ ( 0 , ε ′ ) , δ ∈ ( 0 , δ ′ ) .

Thus,

M μ a , P ( s , ε , δ , f ) = ∞ .

For any Z with μ ( Z ) ≥ 1 − δ . If Z ⊂ ⋃ i Z i , then μ ( ⋃ i Z i ) ≥ 1 − δ . It follows that

∑ i = 1 ∞ M a , P ( s , ε , Z i , f ) = ∞ ,

which implies that M a , P ( s , ε , Z , f ) = ∞ . Hence, P a , P ( ε , Z , f ) ≥ s and P μ a , P ( f ) ≥ s . This shows that P μ a , P ( f ) ≥ P μ a , K P ( f ) .

We shall show the inverse inequality. If s < P μ a , P ( f ) , then there exists ε ′ , δ ′ > 0 such that

inf { P a , P ( ε , Z , f ) : μ ( Z ) ≥ 1 − δ } > s , ∀ ε ∈ ( 0 , ε ′ ) , δ ∈ ( 0 , δ ′ ) .

For any family { Z i } i ≥ 1 with μ ( ⋃ i Z i ) ≥ 1 − δ , we have

P a , P ( ε , ⋃ i Z i , f ) > s .

This implies that

M a , P ( s , ε , ⋃ i Z i , f ) = ∞ .

Thus,

∑ i M a , P ( s , ε , Z i , f ) = ∞ .

Then,

M μ a , P ( s , ε , δ , f ) = ∞ .

Hence,

P μ a , K P ( ε , δ , f ) > s ,

which yields the desired inequality.□

Definition 3.1

[16] For 1 ≤ i ≤ k , we fix open covers { U i } i = 1 k , where U i is a finite cover of X i . For a = ( a 1 , a 2 , … , a k ) , we define the weighted string

U n a ≔ U 1 1 ∩ T 1 − 1 U 2 1 ∩ T 1 − 2 U 3 1 ∩ … ∩ T 1 − ⌈ a 1 n ⌉ − 1 U ⌈ a 1 n ⌉ 1 ∩ τ 1 − 1 U 1 2 ∩ τ 1 − 1 T 2 − 1 U 2 2 ∩ τ 1 − 1 T 2 − 2 U 3 2 ∩ … ∩ τ 1 − 1 T 2 − ⌈ ( a 1 + a 2 ) n ⌉ − 1 U ⌈ ( a 1 + a 2 ) n ⌉ 2 … ∩ τ k − 1 − 1 U 1 k ∩ τ k − 1 − 1 T k − 1 U 2 k ∩ τ k − 1 − 1 T k − 2 U 3 k ∩ … ∩ τ k − 1 − 1 T k − ⌈ ( a 1 + a 2 + … a k ) n ⌉ − 1 U ⌈ ( a 1 + a 2 + … + a k ) n ⌉ k

where U j i ∈ U i , for all 1 ≤ i ≤ k , 1 ≤ j ≤ ⌈ ( a 1 + a 2 + … + a k ) n ⌉

Definition 3.2

Let μ be a Borel probability measure on X 1 . Consider finite open covers { U i } i = 1 k . According to [10, Section 10], the C -structure τ = ( S , ℱ , ξ , η , ψ ) on X 1 generates the Carathéodory dimension of μ and lower and upper Carathéodory capacities of μ specified by the covers { U i } i = 1 k and the map f . Replace the a -weighted Bowen ball B n a ( x , ε ) by the weighted string U n a ; it is routine to give an equal definition of weighted Pesin-Pitskel, weighted lower capacity, and weighted upper capacity topological pressures as follows. We denote them by P μ a ( f , { U i } i = 1 k ) , C P ̲ μ a ( f , { U i } i = 1 k ) , and C P ¯ μ a ( f , { U i } i = 1 k ) , respectively. We have that

(3.1) P μ a ( f , { U i } i = 1 k ) = inf { P Z a ( f , { U i } i = 1 k ) : μ ( Z ) = 1 } , C P ¯ μ a ( f , { U i } i = 1 k ) = lim δ → 0 inf { C P ̲ Z a ( f , { U i } i = 1 k ) : μ ( Z ) ≥ 1 − δ } , C P ¯ μ a ( f , { U i } i = 1 k ) = lim δ → 0 inf { C P ¯ Z a ( f , { U i } i = 1 k ) : μ ( Z ) ≥ 1 − δ } .

It is routine to show that there exist the limits

P μ a ( f ) = def lim diam ( { U i } i = 1 k ) → 0 P μ a ( f , { U i } i = 1 k ) , C P ̲ μ a ( f ) = def lim diam ( { U i } i = 1 k ) → 0 C P ̲ μ a ( f , { U i } i = 1 k ) , C P ¯ μ a ( f ) = def lim diam ( { U i } i = 1 k ) → 0 C P ¯ μ a ( f , { U i } i = 1 k ) .

According to [10, Section 10], the C -structure τ = ( S , ℱ , ξ , η , ψ ) on X 1 , we use the weighted string to define the lower and upper α -Carathéodory pointwise dimensions of μ at x as follows.

Definition 3.3

Given α ∈ R and x ∈ X , we define now the lower and upper α -Carathéodory pointwise dimensions of μ at x by

D ̲ C , μ , α ( x , f , { U i } i = 1 k ) = lim ̲ N → ∞ inf U N a α log μ ( U N a ) − N α + sup y ∈ U N a 1 a 1 S ⌈ a 1 N ⌉ f ( y ) , D ¯ C , μ , α ( x , f , { U i } i = 1 k ) = lim ¯ N → ∞ sup U N a α log μ ( U N a ) − N α + sup y ∈ U N a 1 a 1 S ⌈ a 1 N ⌉ f ( y ) ,

where the infimum and supremum are taken over all strings U N a .

Also, we have the following theorems to estimate the dimension of measure.

Theorem 3.1

[10] Assume that there are a number β ≠ 0 and an interval [ β 1 , β 2 ] such that β ∈ ( β 1 , β 2 ) and for μ -almost every x ∈ X and any α ∈ [ β 1 , β 2 ]

if β > 0 , then D ̲ C , μ , α ( x ) ≥ β , and if β < 0 , then D ¯ C , μ , α ( x ) ≤ β ;
there exists ε ( x ) > 0 such that e − N α + sup y ∈ U N a 1 a 1 S ⌈ a 1 N ⌉ f ( y ) < 1 for any set U ( x , ε ) ∈ ℱ ′ ; moreover, the function ε ( x ) is measurable.

Then, dim C μ ≥ β .

Theorem 3.2

[10] Assume that there are a number β ≠ 0 and an interval [ β 1 , β 2 ] such that β ∈ ( β 1 , β 2 ) and for μ -almost every x ∈ X and any α ∈ [ β 1 , β 2 ]

if β > 0 , then D ¯ C , μ , α ( x ) ≤ β , and if β < 0 , then D ̲ C , μ , a ( x ) ≥ β ;
there exists ε ( x ) > 0 such that e − N α + sup y ∈ U N a 1 a 1 S ⌈ a 1 N ⌉ f ( y ) < 1 for any set U ( x , ε ) ∈ ℱ ′ ; moreover, the function ε ( x ) is measurable;

Then, Cap ¯ C μ ≤ β .

Theorem 3.3

[14] For each ergodic measure μ ∈ ℳ ( X 1 , T 1 ) , we have

lim ε → 0 liminf n → + ∞ − log μ ( B n a ( x , ε ) ) n = lim ε → 0 limsup n → + ∞ − log μ ( B n a ( x , ε ) ) n = h μ a ( T 1 ) ,

for μ -a.e. x ∈ X 1 . When a = ( 1 , 0 , … , 0 ) , the aforementioned result reduces to the Brin-Katok theorem on local entropy [14].

Proposition 3.3

If μ is a Borel probability measure on X 1 invariant under the map f and ergodic, then for every α ∈ R and μ -almost every x ∈ X 1 ,

lim diam ( { U i } i = 1 k ) → 0 D ̲ C , μ , α ( x , f , { U i } i = 1 k ) = lim diam ( { U i } i = 1 k ) → 0 D ¯ C , μ , α ( x , f , { U i } i = 1 k ) = α h μ a ( T 1 ) α − ∫ X 1 f d μ ,

where h μ a ( T 1 ) is the measure-theoretic entropy of T 1 .

Proof

Let { U i } i = 1 k be finite open covers of X i , i = 1 , 2 , … , N and δ ( { U i } i = 1 k ) ≔ min 1 ≤ i ≤ k δ ( U i ) , where δ ( U i ) denotes its Lebesgue number. δ ( { U i } i = 1 k ) → 0 as diam ( { U i } i = 1 k ) → 0 . It is easily seen that for every x ∈ X 1 , if x ∈ U N a , then

B n a x , 1 2 δ ( { U i } i = 1 k ) ⊂ U n a ⊂ B n a ( x , 2 ( { U i } i = 1 k ) ) .

Combining with Theorem 3.3,

(3.2) h μ a ( T 1 ) = lim diam ( { U i } i = 1 k ) → 0 lim ̲ N → ∞ inf U N a − log μ ( U N a ) N = lim diam ( { U i } i = 1 k ) → 0 lim ¯ N → ∞ sup U N a − log μ ( U N a ) N ,

where the infimum and supremum are taken over all strings U for which x ∈ U N a . Let us fix a number ε > 0 . Since f is continuous on X 1 , there exists a number δ > 0 such that ∣ f ( x ) − f ( y ) ∣ ≤ ε for any two points x , y ∈ X 1 with d 1 ( x , y ) ≤ δ . Therefore, if diam ( { U i } i = 1 k ) ≤ δ , then by view of Birkhoff ergodic theorem, we obtain for μ -almost every x ∈ X 1 that

liminf N → ∞ inf U N a sup y ∈ U N a 1 N 1 a 1 S ⌈ a 1 N ⌉ f ( y ) − ∫ X 1 f d μ ≤ ε , limsup N → ∞ sup U N a sup y ∈ U N a 1 N 1 a 1 S ⌈ a 1 N ⌉ f ( y ) − ∫ X 1 f d μ ≤ ε ,

where the infimum and supremum are taken over all strings U for which x ∈ U N a . Since ε is arbitrary, this implies that

(3.3) lim diam ( { U i } i = 1 k ) → 0 liminf N → ∞ inf U N a sup y ∈ U N a 1 N 1 a 1 S ⌈ a 1 N ⌉ f ( y ) = lim diam ( { U i } i = 1 k ) → 0 limsup N → ∞ sup U N a sup y ∈ U N a 1 N 1 a 1 S ⌈ a 1 N ⌉ f ( y ) = ∫ X 1 f d μ .

The desired result follows immediately from (3.2) and (3.3) .□

Proposition 3.4

Let f be a continuous function of a compact metric space X 1 and μ a non-atomic Borel ergodic measure on X 1 . Then,

P μ a ( f ) = C P ̲ μ a ( f ) = C P ¯ μ a ( f ) = h μ a ( T 1 ) + ∫ X 1 f d μ .

Proof

Set h a = h μ a ( f ) ≥ 0 and a = ∫ X f d μ . We first assume that a > 0 . We wish to use Theorems 3.1 and 3.2 to obtain the proper lower bound for P μ a ( f ) and upper bound for C P ¯ μ a ( f ) . To do so, we need to find estimates of D ̲ C , μ , α ( x , f , { U i } i = 1 k ) and D ¯ C , μ , α ( x , f , { U i } i = 1 k ) from below and above, respectively, which do not depend on α .

Fix ε , 0 < ε < a 2 . By Theorem 3.3, one can choose δ > 0 such that for μ -almost every x ∈ X 1 ,

D ̲ C , μ , α ( x , f , { U i } i = 1 k ) ≥ α h α − a − ε .

Note that the function g ( α ) = α h ( α − a ) − 1 − ε is decreasing. Assuming that α varies on the interval [ h + a − ε , h + a ] , we obtain that for μ -almost every x ∈ X 1 ,

D ̲ C , μ , a ( x , f , { U i } i = 1 k ) ≥ h + a − 2 ε .

We conclude, using Theorem 3.1, that P μ a ( f , { U i } i = 1 k ) ≥ h + a − 2 ε , and hence, P μ a ( f , { U i } i = 1 k ) ≥ h + a . Since this holds for every finite open covers { U i } i = 1 k , by (3.1), we obtain that P μ a ( f ) ≥ h + a .

We now show that C P ¯ μ a ( f ) ≤ h + a . Fix ε > 0 . We can choose ξ i = { C 1 1 , … , C p i 1 } be a finite measurable partition of X i for any 1 ≤ i ≤ k with

∑ i = 1 k h μ ∘ τ i − 1 − 1 ( T i , ∨ j = 0 ⌈ ( a 1 + … a j ) n ⌉ − 1 T i − j ξ i ) − h ≤ ε

and U i = { U 1 , … , U p i } a finite open cover of X i of diameter ≤ ε for which C j i ⊂ U j i , j = 1 , … , p i .

By the Birkhoff ergodic theorem for μ -almost every x ∈ X 1 , there exists a number N 1 ( x ) > 0 such that for any n ≥ N 1 ( x ) ,

(3.4) 1 a 1 n S ⌈ a 1 N ⌉ f ( x ) − a ≤ ε .

By the proof of weighted Shannon-McMillan-Breiman theorem [14] for μ -almost every x ∈ X 1 , there exists a number N 2 ( x ) > 0 such that for any n ≥ N 2 ( x ) ,

(3.5) 1 n log μ ( ∨ j = 0 ⌈ ( a 1 + … a j ) n ⌉ − 1 T 1 − j τ i − 1 ξ i ( x ) ) + ∑ i = 1 k a i h μ ∘ τ i − 1 − 1 ( T i , ∨ j = 0 ⌈ ( a 1 + … a j ) n ⌉ − 1 T i − j ξ i ) ≤ ε .

Let Δ be the set of points for which (3.4) and (3.5) hold. Given N > 0 , consider the set Δ N = { x ∈ Δ : N 1 ( x ) ≤ N and N 2 ( x ) ≤ N } . We have that Δ N ⊂ Δ N + 1 and Δ = ∪ N ≥ 0 Δ N Therefore, given δ > 0 , one can find N 0 > 0 for which μ ( Δ N 0 ) ≥ 1 − δ . Fix a number N ≥ N 0 and a point x ∈ Δ N . Let U N a be a string of length m ( U N a ) = N for which x ∈ U n a . It follows from (3.4) that

(3.6) sup y ∈ U N a 1 a 1 N S ⌈ a 1 N ⌉ f ( y ) − a ≤ ε + γ ,

where γ = γ ( U ) ≔ max 1 ≤ i ≤ k sup { ∣ f ( x ) − f ( y ) ∣ : x , y ∈ U j i , 1 ≤ j ≤ p i } . Furthermore, using (3.5), we obtain that

μ ( ∨ j = 0 ⌈ ( a 1 + … a j ) n ⌉ − 1 T 1 − j τ i − 1 ξ i ( x ) ) ≥ exp ( − h − 2 ε ) N .

This implies that the number of elements of the partition ∨ j = 0 ⌈ ( a 1 + … a j ) n ⌉ − 1 T 1 − j τ i − 1 ξ i that have non-empty intersection with the set Δ N does not exceed exp ( h + 2 ε ) N .

To each element C ∨ j = 0 ⌈ ( a 1 + … a j ) n ⌉ − 1 T 1 − j τ i − 1 ξ i of the partition ξ N , we associate a string U N a of length m ( U ) = N for which C ∨ j = 0 ⌈ ( a 1 + … a j ) n ⌉ − 1 T 1 − j τ i − 1 ξ i ⊂ U N a . The collection of such strings consists of at most exp ( h + 2 ε ) N elements that comprise a cover G of Δ N . By (3.5) and (3.6), we obtain that

Λ ( Δ N , f , { U i } i = 1 k , N ) ≤ ∑ U ∈ G exp sup y ∈ U N a 1 a 1 S ⌈ a 1 N ⌉ f ( y ) ≤ exp ( a + h + 3 ε + γ ) N .

Then,

C P ¯ Δ N a ( f , { U i } i = 1 k ) ≤ a + h + 3 ε + γ .

This implies that C P ¯ μ a ( f , { U i } i = 1 k ) ≤ a + h + 3 ε + γ . Passing to the limit as diam { U i } i = 1 k → 0 yields that C P ¯ μ a ( f ) ≤ a + h + 3 ε . It remains to note that ε can be chosen arbitrarily small to conclude that C P ¯ μ a ( f ) ≤ a + h .

In the case a ≤ 0 , let us consider a function ψ = f + C , where C is chosen such that ∫ X ψ d μ > 0 . Note that P μ a ( ψ , { U i } i = 1 k ) = P μ a ( f , { U i } i = 1 k ) + C and C P ¯ μ a ( ψ , { U i } i = 1 k ) = C P ¯ μ a ( f , { U i } i = 1 k ) + C , and the desired result follows.□

Now, we can prove the Theorem 2.1 as follows.

Proof

Employing Proposition 3.4, the following equalities follow

(3.7) P μ a , B ( f ) = C P ̲ μ a ( f ) = C P ¯ μ a ( f ) = h μ a ( T 1 ) + ∫ f d μ .

From the proof of Proposition 3.1, it is easy to check that for any Z ⊂ X and ε > 0 ,

P a , B ( ε , Z , f ) ≤ P a , P ( ε , Z , f ) ≤ C P ¯ a ε 2 , Z , f .

Thus,

P μ a , B ( f ) ≤ P μ a , P ( f ) ≤ C P ¯ μ a ( f ) ,

which together with (3.7) yields

P μ a , P ( f ) = h μ a ( T 1 ) + ∫ f d μ .

Using Proposition 3.2, we obtain

P μ a , K P ( f ) = P μ a , P ( f ) = h μ a ( T 1 ) + ∫ f d μ

and

P μ a , B ( f ) = P μ a , K B ( f ) ≤ C P ̲ μ a , K ( f ) ≤ C P ¯ μ a , K ( f ) ≤ C P ¯ μ a ( f ) .

The equalities in Theorem 2.1 then follows.□

Funding information: This work was supported by the Foundation in higher education institutions of Henan Province, PR China (No.23A110020), and Yali Liang was sponsored by “Chenguang Program” (20CGB09) supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission. Junjie Zhang was supported by Jiangsu Province Postgraduate Research and Innovation Program (No. KYCX23_3300).
Author contributions: Bin Zhang wrote the manuscript, Yali Liang conceived the idea, and Junjie Zhang analyzed the data.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR 119 (1958), 861–864 (in Russian)Search in Google Scholar

[2] A. N. Kolmogorov, Entropy per unit time as a metric invariant of automorphisms, Dokl. Akad. Nauk SSSR 124 (1959), 754–755. (in Russian)Search in Google Scholar

[3] R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. 10.1090/S0002-9947-1965-0175106-9Search in Google Scholar

[4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. 10.1090/S0002-9947-1971-0274707-XSearch in Google Scholar

[5] E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 324–366 (in Russian)Search in Google Scholar

[6] D. Ruelle, Statistical mechanics on a compact set with zv action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 187 (1973), 237–251. 10.2307/1996437Search in Google Scholar

[7] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1975), 937–971. 10.2307/2373682Search in Google Scholar

[8] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136. 10.1090/S0002-9947-1973-0338317-XSearch in Google Scholar

[9] Y. Pesin and B. S. Pitskel, Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl. 18 (1984), 307–318. 10.1007/BF01083692Search in Google Scholar

[10] Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, IL, 1997. 10.7208/chicago/9780226662237.001.0001Search in Google Scholar

[11] D. Ruelle, Thermodynamic formalism, The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia of Mathematics and Its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. Search in Google Scholar

[12] R. Bowen, Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25. 10.1007/BF02684767Search in Google Scholar

[13] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99–107. 10.1017/S0143385700009603Search in Google Scholar

[14] D. Feng and W. Huang, Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016), 411–452. 10.1016/j.matpur.2016.02.016Search in Google Scholar

[15] M. Tsukamoto, New approach to weighted topological entropy and pressure, Ergodic Theory Dynam. Syst. 43 (2023), no. 3, 1004–1034. 10.1017/etds.2021.173Search in Google Scholar

[16] X. Shao, R. Lu, and C. Zhao, Multifractal analysis of weighted local entropies, Chaos Solitons Fractals 96 (2017), 1–7. 10.1016/j.chaos.2017.01.002Search in Google Scholar

Received: 2023-08-31

Revised: 2024-10-21

Accepted: 2024-12-28

Published Online: 2025-04-30

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https://doi.org/10.1515/math-2024-0125

Keywords for this article

measure-theoretic pressure; weighted pressure; packing topological pressure

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