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 covering condition. Given contracting potential $\varphi$ (in sense Liverani-Saussol-Vaienti), we prove there exists unique conformal measure $\nu_\varphi$ and equilibrium state $\mu_\varphi$. Further, quasi-compactness associated transfer operator cocycle exponential decay correlations Our driving is an invertible, ergodic, measure-preserving transformation $\sigma$ on probability space $(\Omega,\mathscr{F},m)$; each $\omega\in\Omega$ associate piecewise-monotone, surjective map $T_\omega:I\to I$. consider general potentials $\varphi_\omega:I\to\mathbb R\cup\{-\infty\}$ such weight function $g_\omega=e^{\varphi_\omega}$ bounded variation. provide several examples our theory. In particular, results apply to linear non-linear including $\beta$-transformations, randomly translated Gauss-Renyi maps, non-uniformly expanding maps as intermittent with branches, large class Lasota-Yorke maps.