Thermodynamic Formalism for Random Weighted Covering Systems

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 $\varphi$ (in the sense of Liverani-Saussol-Vaienti), we prove there exists a unique random conformal measure $\nu_\varphi$ and unique random equilibrium state $\mu_\varphi$. Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for $\mu_\varphi$. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $\sigma$ on a probability space $(\Omega,\mathscr{F},m)$; for each $\omega\in\Omega$ we associate a piecewise-monotone, surjective map $T_\omega:I\to I$. We consider general potentials $\varphi_\omega:I\to\mathbb R\cup\{-\infty\}$ such that the weight function $g_\omega=e^{\varphi_\omega}$ 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.