Thermodynamic Formalism for Random Interval Maps with Holes

TL;DR

This paper develops a thermodynamic formalism for open random interval maps with holes, establishing existence, uniqueness, and statistical properties of invariant measures, and analyzing escape rates and Hausdorff dimension of survivor sets.

Contribution

It introduces a quenched thermodynamic formalism for open random dynamical systems with holes, proving existence and uniqueness of invariant measures and analyzing their statistical properties.

Findings

Existence of a unique random invariant measure supported on the survivor set.

Proven exponential decay of correlations for the invariant measure.

Escape rates and Hausdorff dimension are characterized by the expected pressure.

Abstract

We develop a quenched thermodynamic formalism for open random dynamical systems generated by finitely branched, piecewise-monotone mappings of the interval. The openness refers to the presence of holes in the interval, which terminate trajectories once they enter; the holes may also be random. 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 $T_\omega:I\to I$, and a hole $H_\omega\subset I$; the map $T_\omega$, the random potential $\varphi_\omega$, and the hole $H_\omega$ generate the corresponding open transfer operator $\mathcal{L}_\omega$. For a contracting potential, under a condition on the open random dynamics in the spirit of Liverani--Maume-Deschamps, we prove there exists a unique random probability measure $\nu_\omega$ supported on the survivor set ${X}_{\omega,\infty}$ satisfying $\nu_{\sigma(\omega)}(\mathcal{L}_\omega f)=\lambda_\omega\nu_\omega(f)$. We also prove the existence of a unique random family of functions $q_\omega$ that satisfy $\mathcal{L}_\omega q_\omega=\lambda_\omega q_{\sigma(\omega)}$. These yield an ergodic random invariant measure $\mu=\nu q$ supported on the global survivor set, while $q$ combined with the random closed conformal measure yields a unique random absolutely continuous conditional invariant measure (RACCIM) $\eta$ supported on $I$. We prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for $\mu$. Finally, the escape rates of the random closed conformal measure and the RACCIM $\eta$ coincide, and are given in terms of the expected pressure, as is the Hausdorff dimension of the surviving set $X_{\omega,\infty}$. We provide examples of our general theory, including random $\beta$-transformations and random Lasota-Yorke maps.