Invariant and conditionally invariant measures for random open interval maps with countably many branches

TL;DR

This paper extends thermodynamic formalism to random open interval maps with countably many branches, establishing key spectral properties, decay of correlations, and analyzing escape rates and conditionally invariant measures.

Contribution

It introduces a Ruelle-Perron-Frobenius theorem for such systems under new assumptions, advancing understanding of their statistical properties.

Findings

Proved exponential decay of correlations.

Established spectral properties of the transfer operator.

Analyzed escape rates and conditionally invariant measures.

Abstract

In this paper, building on previous work, we extend the thermodynamic formalism for random open dynamical systems generated by piecewise monotone interval maps with countably many branches. Under summable and contracting assumptions on the potential, we establish the Ruelle-Perron-Frobenius type theorem for the associated random open operator and prove exponential decay of correlations. In addition, we investigate the escape rate for the hole and conditionally invariant measure for the open system.