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Proceedings of the American Mathematical Society

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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.
DOI: https://doi.org/10.1090/S0002-9939-10-10332-3
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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Additional Information

Lluís Alsedà
Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain
MR Author ID: 212847
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

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.