Thermodynamic Formalism for Random Weighted Covering Systems

Abstract

We develop for the first time 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 $$T_\omega :I\rightarrow I$$ . We consider general potentials $$\varphi _\omega :I\rightarrow {\mathbb {R}}\cup \{-\infty \}$$ 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 new examples of linear and non-linear systems including random $$\beta $$ -transformations, randomly translated random $$\beta $$ -transformations, countably branched random Gauss–Renyi maps, random non-uniformly expanding maps (such as intermittent maps and maps with contracting branches) composed with expanding maps, and a large class of random Lasota–Yorke maps.