Minimal sets of fibre-preserving maps in graph bundles

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 $E$.

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