Random iterated function systems with smooth invariant densities

TL;DR

This paper demonstrates that certain random iterated function systems on the interval possess smooth invariant densities, specifically in the class of infinitely differentiable functions, using cone metric contraction techniques.

Contribution

It introduces a novel application of cone metric contraction methods to establish smoothness of invariant densities for random iterated function systems.

Findings

Invariant measure has a $ ext{C}^\infty$ density.

Techniques involve contractions in cone metrics.

Applicable to specific classes of random IFS.

Abstract

We consider some random iterated function systems on the interval and show that the invariant measure has density in $\mathcal{C}^\infty$. To prove this we use some techniques for contractions in cone metrics, applied to the transfer operator.