On the dimension of stationary measures for random piecewise affine interval homeomorphisms

TL;DR

This paper investigates the Hausdorff dimension of stationary measures for a class of random dynamical systems called Alsedà–Misiurewicz systems, showing that for many parameters, these measures are singular with dimension less than one.

Contribution

It proves that for an open set of parameters, the stationary measures of AM-systems are singular with Hausdorff dimension below one, partially answering a longstanding question.

Findings

Stationary measures have Hausdorff dimension less than 1 for many parameters.

These measures are singular, not absolutely continuous.

The results address a question posed in 2014 by Alsedà and Misiurewicz.

Abstract

We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsed\`a--Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM-system has Hausdorff dimension strictly smaller than $1$. In particular, we obtain singularity of these measures, answering partially a question of Alsed\`a and Misiurewicz from 2014.