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

Chinen, Naotsugu

doi:10.1090/S0002-9939-03-06900-4

Circle maps having an infinite $\omega$-limit set which contains a periodic orbit have positive topological entropy
HTML articles powered by AMS MathViewer

by Naotsugu Chinen PDF

Proc. Amer. Math. Soc. 131 (2003), 3547-3551 Request permission

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.

References

Louis Block, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978), no. 3, 576–580. MR 509258, DOI 10.1090/S0002-9939-1978-0509258-X
L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513, DOI 10.1007/BFb0084762
Louis Block, Ethan Coven, Irene Mulvey, and Zbigniew Nitecki, Homoclinic and nonwandering points for maps of the circle, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 521–532. MR 753920, DOI 10.1017/S014338570000211X
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of one-dimensional maps, Mathematics and its Applications, vol. 407, Kluwer Academic Publishers Group, Dordrecht, 1997. Translated from the 1989 Russian original by Sivak, P. Malyshev and D. Malyshev and revised by the authors. MR 1448407, DOI 10.1007/978-94-015-8897-3