Denjoy constructions for fibred homeomorphisms of the torus

TL;DR

This paper constructs quasiperiodically forced circle homeomorphisms with complex transitive but non-minimal dynamics, filling gaps in topological classification and exploring the structure of minimal sets, including Cantor sets and restrictions under SL(2,R)-cocycles.

Contribution

It demonstrates the existence of transitive, non-minimal quasiperiodic circle homeomorphisms with intricate minimal sets, advancing the topological classification of such dynamical systems.

Findings

Existence of transitive but non-minimal dynamics in various cases.

Construction of minimal sets that are Cantor sets with uncountable, nowhere dense intersections.

Restrictions on minimal sets for quasiperiodic SL(2,R)-cocycles.

Abstract

We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincar\'e-like classification for this class of maps of Jaeger-Stark, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that, in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points). We also prove that minimal sets of the later kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the torus either is a union of finitely many continuous curves, or contains at most two points on generic fibres.