Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T13:44:39.903Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotone recurrence relations, their Birkhoff orbits and topological entropy

Published online by Cambridge University Press:  19 September 2008

Sigurd B. Angenent
Sigurd B. Angenent
Affiliation:
University of Wisconsin—Madison Center for the Mathematical Sciences, Wisconsin, USA
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.

A generalization of the class of monotone twistmaps to maps of s1 × RN is proposed. The existence of Birkhoff orbits is studied, and a criterion for positive topological entropy is given. These results are then specialized to the case of monotone twist maps. Finally it is shown that there is a large class of symplectic maps to which the foregoing discussion applies.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 1 , March 1990 , pp. 15 - 41
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[A]Angenent, S. B.. The periodic orbits of an area preserving twistmap. Comm. Math. Phys. 115 (1988), 353–374.Google Scholar
[ALD]Aubry, S. & Le Daeron, P. Y.. The discrete Frenkel-Kontorova model and its generalizations. Physica 8D (1983), 381–422.Google Scholar
[Ba]Bangert, S.. Mather sets for twist maps and geodesies on tori. To appear in Dynamics Reported, Vol. 1.Google Scholar
[Bo]Boyland, S.. Braid types and a topological method of proving positive entropy. Preprint.Google Scholar
[Ch]Chenciner, S.. Bifurcations de points fixes elliptiques II, orbites periodiques et ensembles de Cantor invariantes. Inventiones. Math. 80 (1985), 81–106.CrossRefGoogle Scholar
[Ha1]Hall, S. R.. A topological version of a theorem of Mather on twist maps. Ergod. Th. & Dynam. Sys. 4 (1984), 585–603.CrossRefGoogle Scholar
[Ha2]Hall, S. R.. A topological version of a theorem of Mather's on shadowing in monotone twist maps. Preprint.Google Scholar
[Ka1]Katok, S.. Some remarks on Birkhoff and Mather twist theorems. Ergod. Th. & Dynam. Sys. 2 (1982), 185–194.Google Scholar
[Ka2]Katok, S.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. de l' I.H.E.S. 51 (1980), 137–174.CrossRefGoogle Scholar
[Ma1]Mather, S. N.. More Denjoy minimal sets for area-reserving diffeomorphisms. Comm. Math. Helv. 60 (1985), 508–557.Google Scholar
[Ma2]Mather, S. N.. Dynamics of area preserving mappings. Preprint.Google Scholar
[Wa]Walters, S.. An introduction to ergodic theory. Grad. Texts in Math., Springer-Verlag: New York, 1982.Google Scholar