# Singular stationary measures for random piecewise affine interval homeomorphisms

## TL;DR

This paper demonstrates that certain random systems of two piecewise affine interval homeomorphisms have singular stationary measures, addressing conjectures and questions about their absolute continuity in the context of semigroups of circle homeomorphisms.

## Contribution

It provides the first proof of singular stationary measures for specific random piecewise affine systems, partially confirming a conjecture and advancing understanding of measure regularity.

## Key findings

- Stationary measures are singular for some random systems of two piecewise affine homeomorphisms.
- Addresses a conjecture by Alsedà and Misiurewicz regarding measure singularity.
- Contributes to the question of absolute continuity of stationary measures in semigroups of circle homeomorphisms.

## Abstract

We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsed\`a and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.