Denjoy constructions for fibered homeomorphisms of the torus

• Published in: Transactions of the American Mathematical Society 2009-11, Vol.361 (11), p.5851-5883
• Main Authors: Béguin, F., Crovisier, S., Jäger, Tobias, Le Roux, F.
• Format: Article
• Language: English
• Subjects: Cardinality; Ergodic theory; Exact sciences and technology; Global analysis, analysis on manifolds; Homeomorphism; Integers; Lebesgue measures; Manifolds and cell complexes; Mathematical intervals; Mathematics; Multivariate analysis; Open intervals; Probability and statistics; Roux; Sciences and techniques of general use; Statistics; Topological theorems; Topology; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-09-04914-9

We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincaré-like classification by Jäger and Stark for this class of maps, 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. \par 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). \par We also prove that minimal sets of the latter kind cannot occur when the dynamics are given by the projective action of a quasiperiodic \mbox {SL}(2,\mathbb R)-cocycle. More precisely, we show that for a quasiperiodic \mbox {SL}(2,\mathbb R)-cocycle, any minimal proper subset of the torus either is a union of finitely many continuous curves or contains at most two points on generic fibres.