# Generic invariant measures for iterated systems of interval homeomorphisms

## TL;DR

This paper explores the invariant measures of iterated systems of interval homeomorphisms, showing the density of systems with singular measures and the residual presence of systems with unique singular stationary distributions.

## Contribution

It introduces a complete metric space for such systems with differentiability, proves the density of systems with singular measures, and establishes a residual set with unique singular stationary distributions.

## Key findings

- Almost singular invariant measures are dense in the space.
- A residual set of systems admits a unique singular stationary distribution.
- Dichotomy between singular and absolutely continuous measures is established.

## Abstract

It is known that Iterated Function Systems generated by orientation preserving homeomorphisms of the unit interval admit a unique invariant measure on $(0,1)$. The setup for this result is the positivity of Lyapunov exponents at both fixed points and the minimality of the induced action. With the additional requirement of continuous differentiability of maps on a fixed neighborhood of $\{0,1\}$, we present a metric in the space of such systems, which renders it complete. Using then a classical argument (and an alternative uniqueness proof), we show that almost singular invariant measures are admitted by systems lying densely in the space. This allows us to construct a residual set of systems with unique singular stationary distribution. Dichotomy between singular and absolutely continuous unique measures, is assured by taking a subspace of systems with absolutely continuous maps; the closure of this subspace is where the residual set is found. The article is dedicated to the memory of Professor J\'ozef Myjak.