Invariant densities for random systems of the interval

TL;DR

This paper explicitly constructs invariant densities for random expanding interval systems, covering various examples like tent maps and $eta$-transformations, enhancing understanding of their invariant measures.

Contribution

It provides an explicit method to find all invariant densities for random expanding interval maps, including systems with holes.

Findings

Explicit densities for random tent maps and W-shaped maps

Complete characterization of invariant densities for systems with only expanding maps

Application to systems with holes, like L"uroth maps with a hole

Abstract

For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. In case the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta$-transformations and random L\"uroth maps with a hole.