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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Periodic orbits of large diameter for circle maps


Authors: Lluís Alsedà and Sylvie Ruette
Journal: Proc. Amer. Math. Soc. 138 (2010), 3211-3217
MSC (2010): Primary 37E10; Secondary 37E15
Published electronically: March 25, 2010
MathSciNet review: 2653946
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Abstract: Let $ f$ be a continuous circle map and let $ F$ be a lifting of $ f$. In this paper we study how the existence of a large orbit for $ F$ affects its set of periods. More precisely, we show that, if $ F$ is of degree $ d\geq 1$ and has a periodic orbit of diameter larger than 1, then $ F$ has periodic points of period $ n$ for all integers $ n\geq 1$, and thus so has $ f$. We also give examples showing that this result does not hold when the degree is nonpositive.


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Lluís Alsedà
Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain
Email: alseda@mat.uab.cat

Sylvie Ruette
Affiliation: Laboratoire de Mathématiques, Bâtiment 425, CNRS UMR 8628, Université Paris-Sud 11, 91405 Orsay cedex, France
Email: Sylvie.Ruette@math.u-psud.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10332-3
PII: S 0002-9939(10)10332-3
Received by editor(s): July 24, 2009
Received by editor(s) in revised form: December 12, 2009, and December 15, 2009
Published electronically: March 25, 2010
Additional Notes: This work was partially supported by MEC grant number MTM2008-01486.
Communicated by: Bryna Kra
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.