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Recurrence of non-resonant homeomorphisms on the torus


Author: Rafael Potrie
Journal: Proc. Amer. Math. Soc. 140 (2012), 3973-3981
MSC (2000): Primary 37E45; Secondary 37B20
DOI: https://doi.org/10.1090/S0002-9939-2012-11249-3
Published electronically: March 27, 2012
MathSciNet review: 2944736
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Abstract: We prove that a homeomorphism of the torus homotopic to the identity whose rotation set is reduced to a single totally irrational vector is chain-recurrent. In fact, we show that pseudo-orbits can be chosen with a small number of jumps, in particular, that the non-wandering set is weakly transitive. We give an example showing that the non-wandering set of such a homeomorphism may not be transitive.


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Additional Information

Rafael Potrie
Affiliation: CMAT, Facultad de Ciencias, Universidad de la República, 11400 Montevideo, Uruguay
Address at time of publication: LAGA, Institute Galilee, Université Paris 13, Villetaneuse, France
Email: rpotrie@cmat.edu.uy

DOI: https://doi.org/10.1090/S0002-9939-2012-11249-3
Keywords: Torus homeomorphism, rotation vectors.
Received by editor(s): March 23, 2011
Received by editor(s) in revised form: May 2, 2011, and May 12, 2011
Published electronically: March 27, 2012
Additional Notes: The author was partially supported by ANR Blanc DynNonHyp BLAN08-2$_$313375 and ANII Proyecto FCE2007$_$577.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.