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Quenched and annealed equilibrium states for random Ruelle expanding maps and applications

Manuel Stadlbauer, Paulo Varandas, Xuan Zhang

Abstract

In this paper we describe the spectral properties of semigroups of expanding maps acting on Polish spaces, considering both sequences of transfer operators along infinite compositions of dynamics and integrated transfer operators. We prove that there exists a limiting behaviour for such transfer operators, and that these semigroup actions admit equilibrium states with exponential decay of correlations and several limit theorems. The reformulation of these results in terms of quenched and annealed equilibrium states extend results by Baladi (1997) and Carvalho, Rodrigues & Varandas (2017), where the randomness is driven by a random walk and the phase space is assumed to be compact. Furthermore, we prove that the quenched equilibrium measures vary Hölder continuously and that the annealed equilibrium states can be recovered from the latter. Finally, we give some applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and on the boundary of equilibria.

Quenched and annealed equilibrium states for random Ruelle expanding maps and applications

Abstract

In this paper we describe the spectral properties of semigroups of expanding maps acting on Polish spaces, considering both sequences of transfer operators along infinite compositions of dynamics and integrated transfer operators. We prove that there exists a limiting behaviour for such transfer operators, and that these semigroup actions admit equilibrium states with exponential decay of correlations and several limit theorems. The reformulation of these results in terms of quenched and annealed equilibrium states extend results by Baladi (1997) and Carvalho, Rodrigues & Varandas (2017), where the randomness is driven by a random walk and the phase space is assumed to be compact. Furthermore, we prove that the quenched equilibrium measures vary Hölder continuously and that the annealed equilibrium states can be recovered from the latter. Finally, we give some applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and on the boundary of equilibria.

Paper Structure

This paper contains 14 sections, 20 theorems, 181 equations, 3 figures.

Table of Contents

  1. Introduction and statement of results
  2. Semigroups of Ruelle expanding maps on non-compact spaces
  3. Quotients of Ruelle operators
  4. Contraction in the Vaserstein distance
  5. Conformal measures, quenched exponential decay and continuity
  6. One-sided dynamics
  7. One-sided dynamics
  8. Two-sided compositions
  9. Annealed exponential decay
  10. An almost sure invariance principle
  11. Applications
  12. Non-autonomous conformal iterated function systems
  13. Thermodynamic formalism of semigroup actions
  14. A boundary of equilibria

Key Result

Theorem A

Suppose the Ruelle-expanding semigroup $\mathcal{S}$ is jointly topologically mixing and finitely aperiodic, and that every potential $\varphi_i$ is $\alpha$-Hölder and summable. Then there exist $k_0 \in {\mathbb N}$ and $s \in (0,1)$ such that for all finite words $u,v$ with $|v|\ge k_0$ and $\nu_

Figures (3)

  • Figure 1: Finite aperiodicity
  • Figure 2: Selection of preimages
  • Figure 3: The map $x\mapsto x^\#$

Theorems & Definitions (50)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 3.1
  • ...and 40 more