Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variational principles of relative weighted topological pressures

Zhengyu Yin

TL;DR

The paper develops a relative theory for weighted topological pressure in topological dynamical systems with factor maps, defining P^omega_Z(pi,T,f) on a common factor and proving two variational principles that connect topological pressure with conditional entropies h_mu(T|R) and h_{pi mu}(S|R). The approach extends weighted pressure concepts from Tsukamoto and Feng–Huang and leverages conditional entropy formalism, zero-dimensional principal extensions, and Rohlin–Abramov-type relations to establish both directions of the variational equalities. A key contribution is the weight parameter omega, which unifies relative entropy and weighted pressure in a single framework and yields weighted relative entropy results as corollaries when f=0. The results provide a robust toolkit for relative weighted thermodynamic formalism with potential applications to fractal geometry, self-similar structures, and dynamical systems with common factors.

Abstract

Recently, M. Tsukamoto (New approach to weighted topological entropy and pressure, Ergod. Theory Dyn. Syst. 43 (2023) 1004-1034) used a new approach to define the weighted topological entropy and pressure. Inspired by his ideas, we introduce the relative weighted topological entropy and pressure for factor maps and establish several variational principles. One of these results involves a question raised by D. Feng and W. Huang (Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016) 411-452), whether there is a relative version of the weighted variation principle. In this paper, we try to establish such variational principle. Furthermore, we generalize the Ledrappier and Walters' type relative variational principle to the weighted version.

Variational principles of relative weighted topological pressures

TL;DR

The paper develops a relative theory for weighted topological pressure in topological dynamical systems with factor maps, defining P^omega_Z(pi,T,f) on a common factor and proving two variational principles that connect topological pressure with conditional entropies h_mu(T|R) and h_{pi mu}(S|R). The approach extends weighted pressure concepts from Tsukamoto and Feng–Huang and leverages conditional entropy formalism, zero-dimensional principal extensions, and Rohlin–Abramov-type relations to establish both directions of the variational equalities. A key contribution is the weight parameter omega, which unifies relative entropy and weighted pressure in a single framework and yields weighted relative entropy results as corollaries when f=0. The results provide a robust toolkit for relative weighted thermodynamic formalism with potential applications to fractal geometry, self-similar structures, and dynamical systems with common factors.

Abstract

Recently, M. Tsukamoto (New approach to weighted topological entropy and pressure, Ergod. Theory Dyn. Syst. 43 (2023) 1004-1034) used a new approach to define the weighted topological entropy and pressure. Inspired by his ideas, we introduce the relative weighted topological entropy and pressure for factor maps and establish several variational principles. One of these results involves a question raised by D. Feng and W. Huang (Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016) 411-452), whether there is a relative version of the weighted variation principle. In this paper, we try to establish such variational principle. Furthermore, we generalize the Ledrappier and Walters' type relative variational principle to the weighted version.
Paper Structure (10 sections, 24 theorems, 191 equations)

This paper contains 10 sections, 24 theorems, 191 equations.

Table of Contents

  1. introduction
  2. Relative weighted topological pressure
  3. Relative weighted topological pressures
  4. Conditional metric entropy
  5. Main results
  6. Basic properties
  7. Zero-dimensional principal extension revisited
  8. Variational principles
  9. Proof of one side of variational principles
  10. Proof of the other side of variational principles

Key Result

Lemma 2.1

Let $\pi: X\to Y$ be a quotient (surjective) map between two topological spaces. Then $\pi$ is closed if and only if for any open subset $U$ of $X$, the union of all fibers of $\pi$ contained in $U$ is open.

Theorems & Definitions (50)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • Theorem 2.8
  • ...and 40 more