Hostname: page-component-7bb8b95d7b-dvmhs Total loading time: 0 Render date: 2024-09-18T01:59:59.790Z Has data issue: false hasContentIssue false
  • English
  • Français

An estimate of entropy for toroidal chaos

Published online by Cambridge University Press:  19 September 2008

Jaroslaw Kwapisz
Jaroslaw Kwapisz
Affiliation:
Institute of Mathematics, Warsaw University, Banacha 2, Warsaw 59, Poland
Get access Rights & Permissions [Opens in a new window]

Abstract

For a mapping F:ℝ2→ℝ2 being a lift of an isotopic to the identity homeomorphism of the two-dimensional torus the rotation set ρ(F) consists of limit points of all sequences where xi, ∈ ℝ2 and ni→∞. It is known that if ρ(F) has nonempty interior then h(f) (the topological entropy of ƒ) is positive. We provide an estimate from below of h(f) in terms of ρ(F).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 1 , March 1993 , pp. 123 - 129
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Ab]Abikoff, W.. The Real Analytic Theory of Teichmuller Space. Springer-Verlag, Berlin, 1980.Google Scholar
[Ah]Ahlfors, L.. Lectures on Quasiconformal Mappings, van Nostrand, New York, 1966.Google Scholar
[BGKM]Baesens, C., Guckenheimer, J., Kim, S. & MacKay, R.. Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos. Physica D 49 (1991), 387–475.Google Scholar
[H1]Handel, M.. The entropy of orientation-reversing homeomorphisms of surfaces. Topology 21 (1982), 291–296.CrossRefGoogle Scholar
[H2]Handel, M.. Periodic point free homeomorphisms of . Preprint, 1988.Google Scholar
[F]Franks, J.. Realizing rotation vectors for torus homeomorphisms. Trans. Amer. Math. Soc. 311 (1989), 107–115.CrossRefGoogle Scholar
[FLP]Fathi, A., Laudenbach, F. & Poenaru, V.. Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979).Google Scholar
[KMG]Kim, S., MacKay, R. & Guckenheimer, J.. Resonance for families of torus maps. Nonlinearity 2 (1989), 391–404.CrossRefGoogle Scholar
[LM]Llibre, J. & MacKay, R.. Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Th. & Dynam. Sys. 11 (1991), 115–128.CrossRefGoogle Scholar
[MZ1]Misiurewicz, M. & Ziemian, K.. Rotation sets for maps of tori. J. London Math. Soc. 40 (1989), 490–506.Google Scholar
[MZ2]Misiurewicz, M. & Ziemian, K.. Rotation sets and ergodic measures for torus homeomorphisms. Preprint, 1989.Google Scholar