Minimal sets of fibre-preserving maps in graph bundles

TL;DR

This paper investigates the topological structure of minimal sets in fibre-preserving dynamical systems on graph bundles, revealing conditions under which these sets are nowhere dense or have interior, and describing their fibre structures.

Contribution

It characterizes the structure of minimal sets in fibre-preserving maps on graph bundles, including conditions for when they form sub-bundles and detailed fibre configurations.

Findings

Minimal sets are either nowhere dense or have nonempty interior.

Fibre structures are either Cantor sets, finite sets, or unions of circles.

Conditions are provided for minimal sets to be sub-bundles.

Abstract

Topological structure of minimal sets is studied for a dynamical system $(E,F)$ given by a fibre-preserving, in general non-invertible, continuous selfmap $F$ of a graph bundle $E$. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let $M$ be a minimal set of $F$ with full projection onto the base space $B$ of the bundle. We show that $M$ is nowhere dense or has nonempty interior depending on whether the set of so called endpoints of $M$ is dense in $M$ or is empty. If $M$ is nowhere dense, we prove that either a typical fibre of $M$ is a Cantor set, or there is a positive integer $N$ such that a typical fibre of $M$ has cardinality $N$. If $M$ has nonempty interior we prove that there is a positive integer $m$ such that a typical fibre of $M$, in fact even each fibre of $M$ over a \emph{dense open} set $\mathcal O \subseteq B$, is a disjoint union of $m$ circles. Moreover, we show that each of the fibres of $M$ over $B\setminus \mathcal O$ is a union of circles properly containing a disjoint union of $m$ circles. Surprisingly, some of the circles in such "non-typical" fibres of $M$ may intersect. We also give sufficient conditions for $M$ to be a sub-bundle of $E$.