Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-17T17:07:22.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Birkhoff-Hénon attractors for dissipative perturbations of area-preserving twist maps

Published online by Cambridge University Press:  19 September 2008

Leonardo Mora
Leonardo Mora
Affiliation:
Instituto Venezolano de Investigaciones Cientificas, Departamento de Matemáticas, Apartado 21827, Caracas 1020-A, Venezuela
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that an area-preserving twist map having an invariant curve, can be approximated by a twist map exhibiting a Birkhoff-Hénon attractor. This is done by showing that the invariant curve can be perturbed into a saddle-node cycle with criticalities and by using a recent result reported by Diaz, Rocha and Viana.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 4 , December 1994 , pp. 807 - 815
Copyright
Copyright © Cambridge University Press 1994

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

[B]Birkhoff, G.. Sur quelques courbes fermées remarquables. Bull. Soc. Math. Fr. 60 (1932), 1–26.Google Scholar
[BC]Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73–169.CrossRefGoogle Scholar
[BY]Benedicks, M. and Young, L.. Sinai-Bowen-Ruelle measures for certain Hénon maps. Invent. Math. 112 (1993), 541–576.CrossRefGoogle Scholar
[DRV]Diaz, J., Rocha, J. and Viana, M.. Saddle-node cycles and prevalence of strange attractors. In preparation.Google Scholar
[H1]Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle á des rotations. Publ. Math IHES 49 (1979), 5–234.Google Scholar
[H2]Herman, M.. Sur les courbes invariants par les difféomorphismes de lánneu. Astérisque 103–104 (1983).Google Scholar
[LI]Calvez, P. Le. Étude topologique des applications déviant la verticale. Ensaios Matemáticos SBM 2 (1990).Google Scholar
[L2]Calvez, P. Le. Propiétés des attracteurs de Birkhoff. Ergod. Th. & Dynam. Sys. 8 (1987), 241–310.CrossRefGoogle Scholar
[MV]Mora, L. and Viana, M.. Abundance of strange attractors. Acta Math. 171 (1993), 1–71.Google Scholar
[M]Moser, J.. Stable and Random Motions in Dynamical Systems. Ann. Math. Studies Vol. 77. Princeton University Press: Princeton, 1973.Google Scholar
[N]Newhouse, S.. Quasi-elliptic periodic points in conservative dynamical systems. Amer. J. Math. 99 (1977), 1061–1087.Google Scholar
[NPT]Newhouse, S., Palis, J. and Takens, F.. Bifurcations and stability of families of diffeomorphismes. Publ. Math. IHES 57 (1983), 5–71.CrossRefGoogle Scholar
[S]Smale, S.. On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff. New YorkAcademic of Sciences: 1980. pp 260–266.Google Scholar