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Equilibrium states for non-transitive random open and closed dynamical systems

Part of: Low-dimensional dynamical systems Random dynamical systems Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  13 October 2022

JASON ATNIP Open the ORCID record for JASON ATNIP [Opens in a new window] ,
GARY FROYLAND ,
CECILIA GONZÁLEZ-TOKMAN  and
SANDRO VAIENTI
JASON ATNIP*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
GARY FROYLAND
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: g.froyland@unsw.edu.au)
CECILIA GONZÁLEZ-TOKMAN
Affiliation:
School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia (e-mail: cecilia.gt@uq.edu.au)
SANDRO VAIENTI
Affiliation:
Aix Marseille Université, Université de Toulon, CNRS, CPT, 13009 Marseille, France (e-mail: vaienti@cpt.univ-mrs.fr)
*
e-mail: j.atnip@unsw.edu.au
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Abstract

We prove a random Ruelle–Perron–Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. This theorem provides the existence and uniqueness of random conformal and invariant measures with exponential decay of correlations, and allows us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and the thermodynamic formalism. Applications include open and closed non-transitive random maps, and a connection between Lyapunov exponents and escape rates through random holes. We are also able to treat random intermittent maps with geometric potentials.

Keywords

random dynamical systemsthermodynamic formalismequilibrium statesopen dynamics

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37H15: Multiplicative ergodic theory, Lyapunov exponents 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 10 , October 2023 , pp. 3193 - 3215
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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