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Thermodynamic formalism for random non-uniformly expanding maps

Manuel Stadlbauer, Shintaro Suzuki, Paulo Varandas

Abstract

We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered $C^1$-potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Carathéodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.

Thermodynamic formalism for random non-uniformly expanding maps

Abstract

We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered -potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Carathéodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.

Paper Structure

This paper contains 30 sections, 33 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Setting and statement of the main results
  3. Examples
  4. Random Manneville-Pomeau maps
  5. Random perturbations of non-uniformly expanding maps
  6. Random maps which expand weakly in average
  7. Multidimensional non-uniformly expanding maps with unbounded degree
  8. Intermittent maps with unbounded degree
  9. Random non-uniformly expanding fractals
  10. Preliminaries
  11. Random thermodynamic formalism
  12. Random non-uniform expansion
  13. Birkhoff cones and random Ruelle-Perron-Frobenius operator
  14. Random Ruelle-Perron-Frobenius operator
  15. Invariant cones and normalized densities
  16. ...and 15 more sections

Key Result

Theorem A

Let $f=(f_\omega)_{\omega\in\Omega}$ be a family of $C^1$-local diffeomorphisms satisfying (H0)-(H3) and let $\phi=(\phi_\omega)_{\omega\in\Omega}$ be a potential function in $L^1_X(\Omega, C^1(M))$ satisfying (P). Then there exists a triple $(\lambda, h, \nu)$ consisting of a positive random variab Moreover, the measure $\mu=(\mu_\omega)_\omega$ defined by $\mu_\omega=h_\omega \nu_\omega$ is an $

Theorems & Definitions (69)

  • Remark \oldthetheorem
  • Theorem A
  • Corollary 1
  • Theorem B
  • Theorem C
  • Theorem D
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 59 more