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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)
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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.

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

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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