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Robust nonhyperbolic dynamics and heterodimensional cycles

Published online by Cambridge University Press:  19 September 2008

Lorenzo J. Díaz
Lorenzo J. Díaz
Affiliation:
Dto. Matemática, PUC-Rio, R. Marquês de S. Vicente 225, 22453 Rio de Janeiro, Brazil
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Abstract

We describe an open set A of arcs of diffeomorphisms bifurcating through the creation of heterodimensional cycles for which every diffeomorphism after the bifurcation is nonhyperbolic or unstable. We also prove that generically in A the borning nonwandering set is transitive and local maximal for a full (Lebesgue) set of parameter values.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 2 , April 1995 , pp. 291 - 315
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[BC]Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73–169.Google Scholar
[D]Díaz, L. J.. Bifurcacoes e ciclos heterodimensionais. Thesis IMPA, Rio de Janeiro, Brazil, 1990.Google Scholar
[DR]Díaz, L. J. and Rocha, J.. Non-connected heterodimensional cycles: bifurcation and stability. Nonlinearity 5 (1992), 1315–1341.CrossRefGoogle Scholar
[HP]Hirsch, M. and Pugh, H.. Stable manifolds and hyperbolic sets. In Global Analysis. Proc. Symp. Pure Math. Vol. 14, Amer. Math. Soc: Rhode Island, 1970.Google Scholar
[MV]Mora, L. and Viana, M.. The abundance of strange attractors. Acta Math. 171 (1993), 1–71.CrossRefGoogle Scholar
[N,1]Newhouse, S.. Nondensity of Axiom A(a) on S 2. In Global Analysis. Proc. Symp. Pure Math. Vol. 14, Amer. Math. Soc: Rhode Island, 1970.Google Scholar
[N,2]Newhouse, S.. Diffeomorphisms with infinitely many sinks. Topology 13 (1974), 9–18.Google Scholar
[N,3]Newhouse, S.. The abundance of wild hyperbolic sets and non smooth stable sets for diffeomorphisms. Publ. IHES 50 (1979), 101–151.CrossRefGoogle Scholar
[NP,1]Newhouse, S. and Palis, J.. Hyperbolic nonwandering sets on two dimensional manifolds. In Dynamical Systems, Academic, New York, 1973.Google Scholar
[NP,2]Newhouse, S. and Palis, J.. Bifurcations of Morse—Smale dynamical systems. In Dynamical Systems, Academic, New York, 1973.Google Scholar
[NP,3]Newhouse, S. and Palis, J.. Cycles and bifurcation theory. Asterisque 31 (1978), 44–140.Google Scholar
[P,1]Palis, J. and Takens, F.. A note on Ω-stability. In Global Analysis. Pwc. Symp. Pure Math. Vol. 14, Amer. Math. Soc: Rhode Island, 1970.Google Scholar
[P,2]Palis, J. and Takens, F.. On the C 1 Ω-stability conjecture. Publ. IHES 66 (1988), 211–215.Google Scholar
[PT,1]Palis, J. and Takens, F.. Cycles and the measure of the bifurcation sets for two-dimensional diffeomorphisms. Invent. Math. 92 (1985), 397–422.Google Scholar
[PT,2]Palis, J.. Hyperbolicity and the creation of homoclinic orbits. Ann. Math. 125 (1987), 337–374.CrossRefGoogle Scholar
[PT,3]Palis, J.. The Theory of Homoclinic Bifurcations: Hyperbolicity, Fractional Dimensions and Infinitely Many Attractors. Cambridge University Press: Cambridge, 1993.Google Scholar
[PY]Palis, J. and Yoccoz, J. C.. Homoclinic bifurcations: Large Hausdorff dimension and non-hyperbolic behaviour. Acta Math. 172 (1994), 91–136.Google Scholar
[Sh]Shub, M.. Stabilité global des systemes dynamiques. Asterisque 56 (1978), 1–211.Google Scholar
[St]Steinberg, S.. On the structure of local homeomorphisms of euclidean n-space II. 80 Amer. J. Math. (1958), 623–631.Google Scholar
[T,1]Takens, F.. Partially hyperbolic fixed points. Topology 8 (1971), 133–147.Google Scholar
[T,2]Takens, F.. Homoclinic bifurcations. Proc. Int. Conf. Math. Berkeley (1988), 1229–1236.Google Scholar
[YA]Yorke, J. and Alligood, K.. Cascades of period-doubling bifurcations: a prerequisite for horseshoes. Bull. Am. Math. Soc. (New Series) 9 (1983), 319–322.CrossRefGoogle Scholar