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Equilibrium States for Random Zooming Systems

Rafael A. Bilbao, Marlon Oliveira, Eduardo Santana

TL;DR

The paper extends equilibrium-state theory to random zooming systems by embedding the dynamics in a skew-product and defining random zooming contractions, random zooming/hyperbolic potentials, and random topological pressure. It proves the existence of equilibrium states for random zooming potentials and, under mild assumptions (e.g., ergodic marginal and dense fiber orbits), establishes uniqueness and common support of ergodic equilibrium states; it also shows the closures of random zooming and random hyperbolic potentials coincide, with the null potential included in these classes. The results are illustrated with random Viana maps (allowing critical points) and a non-expanding random zooming example, highlighting the breadth of the framework beyond uniform hyperbolicity. Overall, the work significantly broadens the equilibrium-state theory to random, nonuniform, and critical-set settings, enabling robust analysis of random nonuniformly expanding dynamics.

Abstract

In this work, based on Pinheiro for deterministic systems, we extend the notion of zooming systems to the random context and based on the technique of Arbieto-Matheus-Oliveira we prove the existence of equilibrium states for which we call random zooming potentials, that include the hyperbolic ones, possibly with the presence of a critical set. With a mild condition, we obtain uniqueness. As an example of existence, we have the so-called random Viana maps with critical points. We also prove that the classes of random zooming potentials and random hyperbolic potentials are equivalent and also contain the null potential, giving measures of maximal entropy.

Equilibrium States for Random Zooming Systems

TL;DR

The paper extends equilibrium-state theory to random zooming systems by embedding the dynamics in a skew-product and defining random zooming contractions, random zooming/hyperbolic potentials, and random topological pressure. It proves the existence of equilibrium states for random zooming potentials and, under mild assumptions (e.g., ergodic marginal and dense fiber orbits), establishes uniqueness and common support of ergodic equilibrium states; it also shows the closures of random zooming and random hyperbolic potentials coincide, with the null potential included in these classes. The results are illustrated with random Viana maps (allowing critical points) and a non-expanding random zooming example, highlighting the breadth of the framework beyond uniform hyperbolicity. Overall, the work significantly broadens the equilibrium-state theory to random, nonuniform, and critical-set settings, enabling robust analysis of random nonuniformly expanding dynamics.

Abstract

In this work, based on Pinheiro for deterministic systems, we extend the notion of zooming systems to the random context and based on the technique of Arbieto-Matheus-Oliveira we prove the existence of equilibrium states for which we call random zooming potentials, that include the hyperbolic ones, possibly with the presence of a critical set. With a mild condition, we obtain uniqueness. As an example of existence, we have the so-called random Viana maps with critical points. We also prove that the classes of random zooming potentials and random hyperbolic potentials are equivalent and also contain the null potential, giving measures of maximal entropy.
Paper Structure (16 sections, 10 theorems, 70 equations)

This paper contains 16 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Result
  3. Random Maps and Invariant Measures
  4. Random Zooming Systems
  5. Entropy and Topological Pressure
  6. Topological Pressure and Variational Principle to noncompact set
  7. Random Zooming Potentials
  8. Random Hyperbolic Potentials
  9. Main Result
  10. Proof of Theorem \ref{['A']}
  11. Proof of Corollary \ref{['B']}
  12. Examples
  13. Random Zooming Potentials
  14. The Null Potential
  15. Random Viana maps
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that $(\Omega, \mathbb{P})$ is a Lebesgue space. Then for any $\phi\in \mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^0(M))$ we have Moreover, when $\mathbb{P}$ is ergodic, we can consider the supremum over ergodic measures.

Theorems & Definitions (16)

  • Definition 1
  • Remark 1
  • Theorem 1: Variational Principle
  • Theorem 2
  • Theorem A
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • Corollary 2
  • ...and 6 more