On Transfer Operators and Maps with Random Holes

TL;DR

This paper investigates Markov interval maps with random holes, demonstrating how their transfer operators relate to deterministic systems, and establishing the existence and properties of invariant measures and their dimensions.

Contribution

It introduces a reduction technique linking random open systems to closed deterministic systems, enabling analysis of invariant measures and Hausdorff dimension.

Findings

Existence of a unique absolutely continuous conditionally stationary measure.

Existence of a unique probability equilibrium measure on the survival set.

Analysis of the Hausdorff dimension of the equilibrium measure.

Abstract

We study Markov interval maps with random holes. The holes are not necessarily elements of the Markov partition. Under a suitable, and physically relevant, assumption on the noise, we show that the transfer operator associated with the random open system can be reduced to a transfer operator associated with the closed deterministic system. Exploiting this fact, we show that the random open system admits a unique (meaningful) absolutely continuous conditionally stationary measure. Moreover, we prove the existence of a unique probability equilibrium measure supported on the survival set, and we study its Hausdorff dimension.