Minimal sets of fibre-preserving maps in graph 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 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.

Minimal sets on continua with a dense free interval

“…A partial description of minimal sets is available for fibre-preserving continuous maps on graph bundles [KST14] and for monotone continuous maps on local dendrites [Abd15] and on regular curves [DM21].…”

Minimal direct products

“…fact, one can show (see, e.g., [KST,Proposition 1]) that if (X, T ) is a (not necessarily compact) minimal dynamical system, then there exists an (irrational) rotation R of the circle S 1 such that the direct product (X × S 1 , T × R) is minimal. This result has proven to be useful in the description of minimal sets of fiber-preserving maps in graph bundles (see [KST]).…”

Construction of minimal non-invertible skew-product maps on 2-manifolds

“…During the last decades, much progress has been made in studying minimal subsystems of (M, f ) in the case when M is a low dimensional compact connected manifold, e.g. see [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”