Thermodynamic Formalism for Random Weighted Covering Systems

Abstract

We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Given a random contracting potential $\phi$ (in the sense of Liverani-Saussol-Vaienti), we prove there exists a unique random conformal measure ${\nu }_{\phi }$ and unique random equilibrium state ${\mu }_{\phi }$. Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for ${\mu }_{\phi }$. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $\sigma$ on a probability space $(\Omega ,\mathcal{F},m)$; for each $\omega \in \Omega$ we associate a piecewise-monotone, surjective map $T_{\omega }:I\to I$. We consider general potentials ${\phi }_{\omega }:I\to \mathbb{R}\cup \{-\infty \}$ such that the weight function $g_{\omega }=e^{{\phi }_{\omega }}$ is of bounded variation. We provide several examples of our general theory. In particular, our results apply to linear and non-linear systems including random $\beta$-transformations, randomly translated random $\beta$-transformations, random Gauss-Renyi maps, random non-uniformly expanding maps such as intermittent maps and maps with contracting branches, and a large class of random Lasota-Yorke maps.

77 pages, 3 figures

References in corpus (1)

• Thermodynamic formalism for random non-uniformly expanding maps

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