Denjoy constructions for fibered homeomorphisms of the torus

Béguin, F.; Crovisier, S.; Jäger, Tobias; Le Roux, F.

doi:10.1090/S0002-9947-09-04914-9

Denjoy constructions for fibered homeomorphisms of the torus
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by F. Béguin, S. Crovisier, Tobias Jäger and F. Le Roux PDF

Trans. Amer. Math. Soc. 361 (2009), 5851-5883 Request permission

Abstract:

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.

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 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.

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