Minimal sets of fibre-preserving maps in graph bundles

Abstract

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