Minimal sets of fibre-preserving maps in graph bundles

الأصل · EN

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 dense open set O B, is a disjoint union of m circles. Moreover, we show that each of the fibres of M over B 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.