Equicontinuity of the Hutchinson operator $F$ and sensitivity of $F_-$

TL;DR

This paper investigates the contrasting dynamical behaviors of the Hutchinson operator and its inverse for a specific iterated function system of circle homeomorphisms, revealing equicontinuity and sensitivity properties.

Contribution

It provides a novel example demonstrating that the Hutchinson operator can be equicontinuous while its inverse's operator is sensitive, highlighting nuanced dynamical differences.

Findings

Hutchinson operator of the system is equicontinuous.

Hutchinson operator of the inverse system is sensitive.

The example illustrates contrasting behaviors in dynamical properties.

Abstract

For an iterated function system $ \mathcal{F} = \{ f_1, \dots, f_k \} $ of homeomorphisms on a compact metric space $(X, d)$, write $ \mathcal{F}_-= \{ f_1^{-1}, \dots, f_k^{-1} \} $. The objective of this paper is to illustrate an iterated function system $\mathcal{F}$ of homeomorphisms on the circle that the Hutchinson operator of $\mathcal{F}$ is equicontinuous, but the Hutchinson operator of $\mathcal{F}_-$ is sensitive.