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open dynamics

Ergodic Theory and Dynamical Systems (1)

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Equilibrium states for non-transitive random open and closed dynamical systems
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JASON ATNIP, GARY FROYLAND, CECILIA GONZÁLEZ-TOKMAN, SANDRO VAIENTI
Journal:

Ergodic Theory and Dynamical Systems / Volume 43 / Issue 10 / October 2023

Published online by Cambridge University Press:

13 October 2022, pp. 3193-3215

Print publication:

October 2023
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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.