Typical behaviour of random interval homeomorphisms

TL;DR

This paper investigates the typical behavior of random interval homeomorphisms with positive Lyapunov exponents, showing that generically the stationary measure is singular yet fully supported, without requiring smoothness.

Contribution

It establishes the typical properties of random interval homeomorphisms under minimal smoothness assumptions, answering a question by Alsedà and Misiurewicz.

Findings

Stationary measure is singular with respect to Lebesgue measure

Stationary measure has full support on [0,1]

Results hold under minimal differentiability assumptions

Abstract

We consider the typical behaviour of random dynamical systems of order-preserving interval homeomorphisms with a positive Lyapunov exponent condition at the endpoints. Our study removes any requirement for continuous differentiability save the existence of finite derivatives of the homeomorphisms at the endpoints of the interval. We construct a suitable Baire space structure for this class of systems. Generically within our Baire space, we show that the stationary measure is singular with respect to the Lebesgue measure, but has full support on $[0,1]$. This provides an answer to a question raised by Alsed\`a and Misiurewicz.