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

Author(s): Naotsugu Chinen
Journal: Proc. Amer. Math. Soc. 131 (2003), 3547-3551.
MSC (2000): Primary 37B40, 37E10; Secondary 28D05, 54H20
Posted: February 14, 2003
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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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L. Block, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978), 576-580. MR 81m:58063

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L. Block and W. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer-Verlag, 1992. MR 93g:58091

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L. Block, E. Coven, I. Mulvey and Z. Nitecki, Homoclinic and non-wandering points for maps of the circle, Ergodic Theory Dynam. Systems, 3 (1983), 521-532. MR 86b:58101

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A. Sharkovsky, S. Kolyada, A. Sivak, and V. Fedorenko, Dynamics of one-dimensional maps, Translated from the 1989 Russian original, Math. and its Appl., 407. Kluwer Academic Publishers Group, Dordrecht, 1997. MR 98k:58083

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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: 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
Posted: February 14, 2003
Communicated by: Ronald A. Fintushel
Copyright of article: Copyright 2003, American Mathematical Society

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Naotsugu Chinen, Circle maps having an infinite ƒÖ-limit set which contains a periodic orbit have positive topological entropy, Proc. Amer. Math. Soc. 131 (2003), 3547-3551.

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