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Nonautonomous Dynamical Systems III: Symbolic and Expansive Systems

Zhuo Chen, Jun Jie Miao

TL;DR

This work develops a comprehensive thermodynamic formalism for nonautonomous dynamical systems, focusing on nonautonomous symbolic dynamics and expansive behavior. It establishes pressure-homogeneity results, explicit formulas for pressures when potentials depend on the first coordinate or have strongly bounded variation, and a law of large numbers to derive measure-theoretic pressures under nonautonomous Bernoulli measures. The paper proves the existence of Bowen and packing equilibrium states, constructs natural nonautonomous Bernoulli equilibria, and connects strongly uniformly expansive systems to symbolic subsystems via generators and equisemiconjugacies. Collectively, these results extend classical autonomous thermodynamics to the nonautonomous setting and provide robust tools for analyzing irregular dynamical sets and nonautonomous fractals. The findings have potential applications in dimension theory, fractal geometry, and the study of nonstationary dynamical phenomena.

Abstract

A nonautonomous dynamical system $(\boldsymbol{X},\boldsymbol{T})=\{(X_{k},T_{k})\}_{k=0}^{\infty}$ is a sequence of continuous mappings $T_{k}:X_{k} \to X_{k+1}$ along with a sequence of compact metric spaces $X_{k}$. In this paper, we study the nonautonomous symbolic dynamical systems and nonautonomous expansive dynamical systems. We first study the homogeneous properties of pressures in nonautonomous symbolic systems $(\boldsymbolΣ(\boldsymbol{m}),\boldsymbolσ)$, and we simplify the formulae of Bowen, packing, lower and upper topological pressures for potentials $\boldsymbol{f}=\{f_{k} \in C(Σ_{k}^{\infty}(\boldsymbol{m}),\mathbb{R})\}_{k=0}^{\infty}$ with strongly bounded variation. Then we apply a law of large numbers to obtain the formulae for the lower and upper measure-theoretic pressures with respect to nonautonomous Bernoulli measures and obtain Bowen equilibrium states and packing equilibrium states for potentials in nonautonomous symbolic systems. Finally, we study the generators in nonautonomous expansive systems $(\boldsymbol{X},\boldsymbol{T})$, and we obtain that $(\boldsymbol{X},\boldsymbol{T})$ is expansive if and only if it has a generator. Moreover, strongly uniformly expansive $(\boldsymbol{X},\boldsymbol{T})$ is equisemiconjugate to a subsystem of the nonautonomous symbolic dynamical system.

Nonautonomous Dynamical Systems III: Symbolic and Expansive Systems

TL;DR

This work develops a comprehensive thermodynamic formalism for nonautonomous dynamical systems, focusing on nonautonomous symbolic dynamics and expansive behavior. It establishes pressure-homogeneity results, explicit formulas for pressures when potentials depend on the first coordinate or have strongly bounded variation, and a law of large numbers to derive measure-theoretic pressures under nonautonomous Bernoulli measures. The paper proves the existence of Bowen and packing equilibrium states, constructs natural nonautonomous Bernoulli equilibria, and connects strongly uniformly expansive systems to symbolic subsystems via generators and equisemiconjugacies. Collectively, these results extend classical autonomous thermodynamics to the nonautonomous setting and provide robust tools for analyzing irregular dynamical sets and nonautonomous fractals. The findings have potential applications in dimension theory, fractal geometry, and the study of nonstationary dynamical phenomena.

Abstract

A nonautonomous dynamical system is a sequence of continuous mappings along with a sequence of compact metric spaces . In this paper, we study the nonautonomous symbolic dynamical systems and nonautonomous expansive dynamical systems. We first study the homogeneous properties of pressures in nonautonomous symbolic systems , and we simplify the formulae of Bowen, packing, lower and upper topological pressures for potentials with strongly bounded variation. Then we apply a law of large numbers to obtain the formulae for the lower and upper measure-theoretic pressures with respect to nonautonomous Bernoulli measures and obtain Bowen equilibrium states and packing equilibrium states for potentials in nonautonomous symbolic systems. Finally, we study the generators in nonautonomous expansive systems , and we obtain that is expansive if and only if it has a generator. Moreover, strongly uniformly expansive is equisemiconjugate to a subsystem of the nonautonomous symbolic dynamical system.

Paper Structure

This paper contains 25 sections, 46 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Nonautonomous dynamics and pressures
  3. Nonautonomous symbolic dynamical systems
  4. Main Conclusions
  5. Topological pressures of nonautonomous shifts
  6. Measure-theoretic pressures
  7. Equilibrium states and Gibbs states
  8. Expansive nonautonomous dynamical systems
  9. Topological Pressures and Entropies
  10. Bowen metrics and Bowen balls
  11. Lower and upper topological pressures and entropies
  12. Bowen pressures and entropies
  13. Packing pressures and entropies
  14. Topological Pressures of Symbolic Dynamical Systems
  15. Equivalent formulations of topological pressures
  16. ...and 10 more sections

Key Result

Theorem 1

Assume that $(\boldsymbol{X},\boldsymbol{T})$ is strongly uniformly expansive with expansive constant $\delta>0$. Then there exists a sequence $\boldsymbol{\mathscr{U}}=\{\mathscr{U}_{k}\}_{k=0}^{\infty}$ of open covers $\mathscr{U}_{k}$ of $X_{k}$ with a Lebesgue number such that where $\#^{\mathrm{cov}}(\mathscr{A})$ denotes the minimal cardinality of subcovers of $\mathscr{A}$.

Theorems & Definitions (85)

  • Theorem : Kawan2015
  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Example 2.1
  • ...and 75 more