Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-09T20:18:13.238Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic isles in families of area-preserving maps

Published online by Cambridge University Press:  15 September 2008

P. DUARTE
P. DUARTE*
Affiliation:
CMAF, Faculdade de Ciências da Universidade de Lisboa, Portugal (email: pduarte@ptmat.fc.ul.pt)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that every one-parameter family of area-preserving maps unfolding a homoclinic tangency has a sequence of parameter intervals, approaching the bifurcation parameter, where the dynamics exhibits wild hyperbolic sets accumulated by elliptic isles. This is a parametric conservative analogue of a famous theorem of Newhouse on the abundance of wild hyperbolic sets.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1781 - 1813
Copyright
Copyright © 2008 Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

[1] Devaney, R. and Nitecki, Z.. Shift automorphisms in Hénon mapping. Commun. Math. Phys. 67 (1979), 137–146.CrossRefGoogle Scholar
[2] Duarte, P.. Plenty of elliptic islands for the standard family of area preserving maps. Ann. Inst. H. Poincare 11(4) (1994), 359–409.Google Scholar
[3] Duarte, P.. Persistent homoclinic tangencies for conservative maps near the identity. Ergod. Th. & Dynam. Sys. 20(2) (2002), 393–438.Google Scholar
[4] Duarte, P.. Abundance of elliptic isles at conservative bifurcations. Dyn. Stab. Syst. 14(4) (1999), 339–356.CrossRefGoogle Scholar
[5] Fontich, E. and Simó, C.. Invariant manifolds for near identity differentiable maps and splitting of separatrices. Ergod. Th. & Dynam. Sys. 10 (1990), 319–346.Google Scholar
[6] Fontich, E. and Simó, C.. The splitting of separatrices for analytic diffeomorphisms. Ergod. Th. & Dynam. Sys. 10 (1990), 295–318.Google Scholar
[7] Gelfreich, V. and Sauzin, D.. Borel summation and splitting of separatrices for the Hénon map. Ann. Inst. Fourier (Grenoble) 51(2) (2001), 513–567.Google Scholar
[8] Gonchenko, S. V., Shilnikov, L. P. and Turaev, D. V.. On the existence of Newhouse regions in a neighbourhood of systems with a structurally unstable Poincare homoclinic curve (the higher-dimensional case). Russ. Acad. Sci. Doklady Mat. 47(2) (1993), 268–273.Google Scholar
[9] Mora, L. and Romero, N.. Moser’s invariant curves and homoclinic bifurcations. Dyn. Systems Appl. 6 (1997), 29–42.Google Scholar
[10] Moreira, G.. Stable intersections of Cantor sets and homoclinic bifurcations. Ann. Inst. H. Poincare 13(6) (1996), 741–781.Google Scholar
[11] Moser, J.. The analytic invariants of an area preserving map near a hyperbolic fixed point. Comm. Pure Appl. Math. 9 (1956), 673–692.Google Scholar
[12] Newhouse, S.. Quasi-elliptic periodic points in conservative dynamical systems. Amer. J. Math. 99(5) (1977), 1061–1067.Google Scholar
[13] Newhouse, S.. Non density of Axiom A(a) on S 2. Proc. Amer. Math. Soc. Symp. Pure Math. 14 (1970), 191–202.CrossRefGoogle Scholar
[14] Newhouse, S.. Diffeomorphisms with infinitely many sinks. Topology 13 (1974), 9–18.CrossRefGoogle Scholar
[15] Newhouse, S.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101–151.Google Scholar
[16] Palis, J. and Takens, F.. Homoclinic Bifurcations and Sensitive Chaotic Dynamics. Cambridge University Press, Cambridge, 1993.Google Scholar
[17] Robinson, C.. Bifurcation to infinitely many sinks. Commun. Math. Phys. 90 (1983), 433–459.Google Scholar
[18] Shub, M.. Global Stability of Dynamical Systems. Springer, Berlin, 1978.Google Scholar
[19] Siegel, C. and Moser, J.. Lectures on Celestial Mechanics. Springer, Berlin, 1971.Google Scholar
[20] Turaev, D.. Newhouse regions in conservative maps. WIAS Preprint, 1998, unpublished.Google Scholar