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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Denjoy constructions for fibered homeomorphisms of the torus


Authors: F. Béguin, S. Crovisier, Tobias Jäger and F. Le Roux
Journal: Trans. Amer. Math. Soc. 361 (2009), 5851-5883
MSC (2000): Primary 37E10; Secondary 37E30, 37E45, 37C55
Published electronically: June 11, 2009
MathSciNet review: 2529917
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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 SL$ (2,\mathbb{R})$-cocycle. More precisely, we show that for a quasiperiodic SL$ (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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Additional Information

F. Béguin
Affiliation: Laboratoire de mathématiques (UMR 8628), Université Paris Sud, 91405 Orsay Cedex, France

S. Crovisier
Affiliation: CNRS et LAGA (UMR 7539), Université Paris 13, Avenue J.-B. Clément, 93430 Villetaneuse, France

Tobias Jäger
Affiliation: Collège de France, 3 rue d’ulm, 75005 Paris, France

F. Le Roux
Affiliation: Laboratoire de mathématiques (UMR 8628), Université Paris Sud, 91405 Orsay Cedex, France

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04914-9
PII: S 0002-9947(09)04914-9
Received by editor(s): September 12, 2007
Published electronically: June 11, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.