A construction of almost automorphic minimal sets

TL;DR

This paper presents a general method for constructing complex minimal sets in dynamical systems by expanding points into segments, enabling the creation of diverse unusual dynamics including almost automorphic and non-distal systems.

Contribution

It introduces a novel procedure to generate topological extensions of skew product maps, producing new examples of minimal sets with intricate dynamical properties.

Findings

Constructed examples of almost automorphic minimal sets.

Produced non-distal homeomorphisms of the torus.

Generated minimal sets of quasiperiodically forced interval maps.

Abstract

We describe a general procedure to construct topological extensions of given skew product maps with one-dimensional fibres, by blowing up a countable number of single points to vertical segments. This allows to produce various examples of unusual dynamics, including almost automorphic minimal sets of almost periodically forced systems, point-distal but non-distal homeomorpisms of the torus (as first constructed by Rees) or minimal sets of quasiperiodically forced interval maps which are not filled-in.