Density of fiberwise orbits in minimal iterated function systems on the circle

TL;DR

This paper investigates the minimality properties of fiberwise orbits in minimal iterated function systems on the circle, demonstrating that such minimality is prevalent and persists under perturbations for systems generated by orientation-preserving homeomorphisms.

Contribution

It establishes the generic minimality of fiberwise orbits in minimal IFSs on the circle and provides new examples showing stability of this behavior under perturbations.

Findings

Minimality of fiberwise orbits holds for most IFSs on the circle.

This minimality persists under small perturbations of the generators.

New examples of IFSs with stable minimality behavior are provided.

Abstract

We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the circle. We provide new examples of iterated functions systems where this behavior persists under perturbation of the generators.