Hostname: page-component-84b7d79bbc-fnpn6 Total loading time: 0 Render date: 2024-07-29T07:20:29.944Z Has data issue: false hasContentIssue false
  • English
  • Français

The speed interval: a rotation algorithm for endomorphisms of the circle

Published online by Cambridge University Press:  19 September 2008

Jan Barkmeijer
Jan Barkmeijer
Affiliation:
Mathematisch Instituut, Postbus 800, 9700 AV Groningen, The Netherlands
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let f be a continuous map of the circle into itself of degree one. We introduce the notion of rotation algorithms. One of these algorithms associates each z ∈ S1 with an interval, the so-called speed interval S(z, f), which is contained in the rotation interval ρ(f) of f. In contrast with the rotation set ρ(z, f), the interval S(z, f) sometimes allows us to ascertain that ρ(f) is non-degenerate, by using only finitely many elements of {fn (z) | n ≥ 0}. We further show that all choices for ρ(z, f) and S(z, f) occur, for certain z ∈ S1 provided that ρ(z, f) ⊂ S(z, f) ⊂ ρ(f).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 1 , March 1988 , pp. 17 - 33
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Alsedà, L., Llibre, J., Misiurewicz, M. & Simó, C.. Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point. Ergod. Th. & Dynam. Sys. 5 (1985), 501–517.CrossRefGoogle Scholar
[2]Alseda, L. & Llibre, J.. A note on the set of periods for continuous maps of the circle which have degree one. Proc. Amer. Math. Soc. 93 (1985), no. 1, 133–139.CrossRefGoogle Scholar
[3]Arnol'd, V. I.. Small denominators I: Mappings of the circumference onto itself. Amer. Math. Soc. Transl., Ser. 2, 46 (1965), 213–284.Google Scholar
[4]Bamon, R., Malta, I. P., Pacifico, M. J. & Takens, F.. Rotation intervals of endomorphisms of the circle. Ergod. Th. & Dynam. Sys. 4 (1984), 493–498.CrossRefGoogle Scholar
[5]Boyland, P. L.. Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals. Commun. Math. Phys. 106 (?1986), 353–381.CrossRefGoogle Scholar
[6]Chenciner, A., Gambaudo, J.-M. & Tresser, Ch.. A remark on degree one endomorphisms of the circle. C.R. Acad. Sc. paris, t. 299, Séar. I, no. 5 (1984).Google Scholar
[7]Ito, R.. Rotation sets are closed. Math. Proc. Camb. Phil. Soc. 89 (1981), 107–111.CrossRefGoogle Scholar
[8]Misiurewicz, M.. Periodic points of maps of degree one of a circle. Ergod. Th. & Dynam. Sys. 2 (1982), 221–227.CrossRefGoogle Scholar
[9]Newhouse, S., Palis, J. & Takens, F.. Bifurcations and stability of families of diffeomorphisms. Publ. Math. I.H.E.S., 57 (1983), 5–71.CrossRefGoogle Scholar
[10]Poincare, H.. Œuvres completes, t.l, Gauthier-Villars (1952), 137–158.Google Scholar