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Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

Published online by Cambridge University Press:  19 September 2008

Lluís Alsedà ,
Jaume Llibre ,
Michał Misiurewicz  and
Carles Simó
Lluís Alsedà
Affiliation:
Seccio de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Departament de Teoria Econòmica, Facultat de Ciències Econòmiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain
Jaume Llibre
Affiliation:
Seccio de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain
Michał Misiurewicz
Affiliation:
Instytut Matematyki, Uniwersytet Warszawski, Pałac Kultury i Nauki IX p., 00–901 Warszawa, Poland
Carles Simó
Affiliation:
Facultat de Matemàtiques, Universitat de Barcelona, Barcelona, Spain
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Abstract

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Let f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 4 , December 1985 , pp. 501 - 517
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Block, L., Coven, E. & Nitecki, Z.. Minimizing topological entropy for maps of the circle. Ergod. Th. & Dynam. Sys. 1 (1981), 145–149.CrossRefGoogle Scholar
[2]Block, L., Guckenheimer, J., Misiurewicz, M. & Young, L. S.. Periodic points and topological entropy of one dimensional maps. Springer Lect. Notes in Math. 819 (1980), 18–39.CrossRefGoogle Scholar
[3]Denker, M., Grillenberger, C. & Sigmund, K.. Ergodic Theory on Compact Spaces. Lect. Notes in Math. 527. Springer, 1976.Google Scholar
[4]Jonker, L. & Rand, D.. The periodic orbits and entropy of certain maps of the unit interval. J. London Math. Soc. 22 (1980), 175–181.CrossRefGoogle Scholar
[5]Misiurewicz, M.. Periodic points of maps of degree one of a circle. Ergod. Th. & Dynam. Sys. 2 (1982), 221–227.CrossRefGoogle Scholar
[6]Misiurewicz, M.. Twist sets for maps of the circle. Preprint, Warsaw, 1983.Google Scholar