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

Part of: Random dynamical systems Dynamical systems with hyperbolic behavior Ergodic theory Special classes of linear operators Smooth dynamical systems: general theory

Published online by Cambridge University Press:  09 September 2022

MANUEL STADLBAUER Open the ORCID record for MANUEL STADLBAUER [Opens in a new window] ,
PAULO VARANDAS Open the ORCID record for PAULO VARANDAS [Opens in a new window]  and
XUAN ZHANG Open the ORCID record for XUAN ZHANG [Opens in a new window]
MANUEL STADLBAUER*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-909, RJ, Brazil
PAULO VARANDAS
Affiliation:
CMUP, Faculdade de Ciências, Universidade do Porto, Porto 4169-007, Portugal Departamento de Matemática, Universidade Federal da Bahia, Salvador 40170-115, BA, Brazil (e-mail: paulo.varandas@ufba.br)
XUAN ZHANG
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo 05508-090, SP, Brazil (e-mail: xuan@ime.usp.br)
*
e-mail: manuel@im.ufrj.br
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Abstract

We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi [Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Comm. Math. Phys. 186 (1997), 671–700] and Carvalho et al [Semigroup actions of expanding maps. J. Stat. Phys. 116(1) (2017), 114–136], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.

Keywords

random dynamical systemsquenched and annealed equilibrium statesnon-autonomous dynamical systemssemigroup actions

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37H05: Foundations, general theory of cocycles, algebraic ergodic theory 47B80: Random operators 37A50: Relations with probability theory and stochastic processes
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 9 , September 2023 , pp. 3150 - 3192
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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