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

 

Circle maps having an infinite $\omega$-limit set which contains a periodic orbit have positive topological entropy


Author: Naotsugu Chinen
Journal: Proc. Amer. Math. Soc. 131 (2003), 3547-3551
MSC (2000): Primary 37B40, 37E10; Secondary 28D05, 54H20
Published electronically: February 14, 2003
MathSciNet review: 1991767
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Abstract: Let $f$ be a continuous map from the circle to itself. The main result of this paper is that the topological entropy of $f$ is positive if and only if $f$ has an infinite $\omega$-limit set which contains a periodic orbit.


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Additional Information

Naotsugu Chinen
Affiliation: Institute of Mathematics, University of Tsukuba, Ibraki 305-8571, Japan
Email: naochin@math.tsukuba.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06900-4
PII: S 0002-9939(03)06900-4
Keywords: $\omega$-limit set, circle, unstable set, homoclinic point, nonwandering point, topological entropy
Received by editor(s): April 15, 2002
Received by editor(s) in revised form: June 24, 2002
Published electronically: February 14, 2003
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2003 American Mathematical Society