Ergodicity of iterated function systems via minimality on the hyper spaces

TL;DR

This paper establishes a sufficient condition for the ergodicity of Lebesgue measure in iterated function systems of diffeomorphisms by analyzing minimality on hyper-spaces, extending to quasi-invariant measures.

Contribution

It introduces a new notion of minimality for induced IFSs on hyper-spaces that guarantees ergodicity, removing the need for $C^1$ regularity in certain cases.

Findings

Ergodicity of Lebesgue measure is proven for several systems.

Minimality on hyper-spaces implies ergodicity of the original IFS.

C^1 regularity is shown to be unnecessary for ergodicity with quasi-invariant measures.

Abstract

We give a sufficient condition for the ergodicity of the Lebesgue measure for an iterated function system of diffeomorphisms. This is done via the induced iterated function system on the space of continuum (which is called hyper-space). We introduce a notion of minimality for induced IFSs which implies that the Lebesgue measure is ergodic for the original IFS. Here, to beginning, the required regularity is $C^1$. However, it is proven that the $C^1$-regularity is a redundant condition to prove ergodicity with respect to the class of quasi-invariant measures. As a consequence of mentioned results, we obtain ergodicity with respect to Lebesgue measure for several systems.