Minimal Strong Foliations in Skew-products of Iterated Function Systems

TL;DR

This paper investigates the minimality of strong foliations in skew-product systems over full shifts, introducing new criteria, and demonstrating the prevalence of minimal foliations in certain settings through analysis of iterated function systems.

Contribution

It introduces a new criterion for foliation minimality, proves minimality is generic for circle fiber systems, and explores stability and approximation of transitive IFS.

Findings

Strong foliations are minimal for an open dense set of systems.

Examples show cases where foliations are not minimal.

Transitive IFS can be approximated by minimal, ergodic systems.

Abstract

We study locally constant skew-product maps over full shifts of finite symbols with arbitrary compact metric spaces as fiber spaces. We introduce a new criterion to determine the density of leaves of the strong unstable (and strong stable) foliation, that is, for its minimality. When the fiber space is a circle, we show that both strong foliations are minimal for an open and dense set of robust transitive skew-products. We provide examples where either one foliation is minimal or neither is minimal. Our approach involves investigating the dynamics of the associated iterated function system (IFS). We establish the asymptotic stability of the phase space of the IFS when it is a strict attractor of the system. We also show that any transitive IFS consisting of circle diffeomorphisms that preserve orientation can be approximated by a robust forward and backward minimal, expanding, and ergodic (with respect to Lebesgue) IFS. Lastly, we provide examples of smooth robust transitive IFSs where either the forward or the backward minimal fails, or both.