Invariant measures for random expanding on average Saussol maps

TL;DR

This paper establishes the existence and finiteness of random absolutely continuous invariant measures for higher-dimensional random expanding Saussol maps, extending previous work to a broader class of maps using a random Lasota-Yorke inequality.

Contribution

It introduces a new approach for proving the existence of invariant measures for higher-dimensional random maps, generalizing prior results to a wider class of systems.

Findings

Finite number of ergodic skew product ACIPs established

Upper bounds for the number of ergodic ACIPs provided

Extension of results to higher-dimensional random maps

Abstract

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota-Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in BGT on admissible random Jab{\l}o\'nski maps to a more general class of higher dimensional random maps.