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.

我们首次为随机动力系统开发了一种淬火热力学形式，该系统由满足随机覆盖条件的区间的可数分支、分段单调映射生成。给定一个随机收缩势\(\varphi \)（在 Liverani-Saussol-Vaienti 的意义上），我们证明存在唯一的随机共形测度\(\nu _\varphi \)和唯一的随机平衡状态\(\mu _\varphi \)。此外，我们证明了相关转移算子 cocycle 的准紧性和\(\mu _\varphi \)相关性的指数衰减。我们的随机驱动是由概率空间上的可逆的、遍历的、保持度量的变换\(\sigma \)生成的\((\Omega ,{\mathscr {F}},m)\) ; 对于每个\(\omega \in \Omega \)我们关联一个分段单调的满射映射\(T_\omega :I\rightarrow I\)。我们考虑一般势\(\varphi _\omega :I\rightarrow {\mathbb {R}}\cup \{-\infty \}\)使得权重函数\(g_\omega =e^{\varphi _ \omega }\)是有界变化的。我们提供了我们一般理论的几个例子。特别是，我们的结果适用于线性和非线性系统的新示例，包括随机\(\beta \)变换、随机平移的随机\(\beta \)-transformations，可数分支随机高斯-Renyi映射，由扩展映射组成的随机非均匀扩展映射（如间歇映射和收缩分支的映射），以及一大类随机Lasota-Yorke映射。