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Newhouse phenomenon and homoclinic classes

Published online by Cambridge University Press:  17 November 2010

JIAGANG YANG
JIAGANG YANG*
Affiliation:
Instituto de Matemática, Universidade Federal Fluminense (UFF), Rio de Janeiro, Brazil (email: [email protected])
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Abstract

We show that for a C1 generic subset of diffeomorphisms far from homoclinic tangencies, any infinite sequence of sinks or sources must accumulate on a homoclinic class of some saddle point with codimension one.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011 , pp. 1537 - 1562
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Abdenur, F., Bonatti, C. and Crovisier, S.. Global dominated splittings and the C 1 Newhouse phenomenon. Proc. Amer. Math. Soc. 134 (2006), 2229–2237.CrossRefGoogle Scholar
[2]Abraham, R. and Smale, S.. Nongenericity of Ω-stability. Proc. Symp. Pure Math. AMS 14 (1970), 5–8.CrossRefGoogle Scholar
[3]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1–235.Google Scholar
[4]Arnaud, M.-C.. Creation de connexions en topologie C 1. Ergod. Th. & Dynam. Sys. 31 (2001), 339–381.CrossRefGoogle Scholar
[5]Bonatti, C.. The global dynamics of C1-generic diffeomorphisms or flows. Second Latin American Congress of Mathematicians. Cancun, Mexico, 2004.Google Scholar
[6]Bonatti, C. and Crovisier, S.. Recurrence et généricite. Invent. Math. 158 (2004), 33–104.CrossRefGoogle Scholar
[7]Bonatti, C. and Díaz, L. J.. Nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143 (1996), 357–396.CrossRefGoogle Scholar
[8]Bonatti, C. and Díaz, L. J.. Connexions hétérocliniques et généricite d’une infinité de puits ou de sources. Ann. Sci. Éc. Norm. Supér. (4) 32 (1999), 135–150.CrossRefGoogle Scholar
[9]Bonatti, C. and Díaz, L. J.. On maximal transitive sets of generic diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 19 (2002), 113–123.Google Scholar
[10]Bonatti, C., Diaz, L. and Pujals, E.. A C 1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158 (2003), 355–418.CrossRefGoogle Scholar
[11]Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics beyond uniform hyperbolic. A Global Geometric and Probabilistic Perspective, Mathematical Physics, III (Encyclopaedia of Mathematical Sciences, 102). Springer, Berlin, 2005.Google Scholar
[12]Bonatti, C., Gan, S. and Wen, L.. On the existence of non-trivial homoclinic classes. Ergod. Th. & Dynam. Sys. 27 (2007), 1473–1508.CrossRefGoogle Scholar
[13]Bonatti, C., Gourmelon, N. and Vivier, T.. Perturbation of the derivative along periodic orbits. Ergod. Th. & Dynam. Sys. 26 (2006), 1307–1337.CrossRefGoogle Scholar
[14]Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.CrossRefGoogle Scholar
[15]Conley, C.. Isolated Invariant Sets and Morse Index (CBM Regional Conference Series in Mathematics, 38). American Mathematical Society, Providence, RI, 1978.CrossRefGoogle Scholar
[16]Crovisier, S.. Periodic orbits and chain transitive sets of C 1 diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 104 (2006), 87–141.CrossRefGoogle Scholar
[17]Crovisier, S.. Birth of homoclinic intersections: a model for the central dynamics of partial hyperbolic systems. Preprint, 2006.Google Scholar
[18]Díaz, L. J.. Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations. Nonlinearity 8 (1995), 693–713.CrossRefGoogle Scholar
[19]Díaz, L. J.. Robust nonhyperbolic dynamics and heterodimensional cycles. Ergod. Th. & Dynam. Sys. 15 (1995), 291–315.CrossRefGoogle Scholar
[20]Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1977), 301–308.CrossRefGoogle Scholar
[21]Gan, S.. Private communication.Google Scholar
[22]Gan, S.. Another proof for C 1 stability conjecture for flows. Sci. China Ser. A 41(10) (1998), 1076–1082.CrossRefGoogle Scholar
[23]Gan, S.. The star systems X * and a proof of the C 1Omega-stability conjecture for flows. J. Differential Equations 163 (2000), 1–17.CrossRefGoogle Scholar
[24]Gan, S. and Wen, L.. Heteroclinic cycles and homoclinic closures for generic diffeomorphisms. J. Dynam. Differential Equations 15 (2003), 451–471.CrossRefGoogle Scholar
[25]Gan, S. and Wen, L.. Nonsingular star flows satisfy Axiom M and the nocycle condition. Invent. Math. 164 (2006), 279–315.CrossRefGoogle Scholar
[26]Gourmelon, N.. Adapted metrics for dominated splitting. Ergod. Th. & Dynam. Sys. 27 (2007), 1839–1849.CrossRefGoogle Scholar
[27]Hayashi, S.. Connecting invariant manifolds and the solution of the C 1 stability and Ω-stable conjecture for flows. Ann. of Math. (2) 145 (1997), 81–137.CrossRefGoogle Scholar
[28]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, New York, 1977.CrossRefGoogle Scholar
[29]Liao, S.. On the stability conjecture. Chinese Ann. Math. 1 (1980), 9–30 (in English).Google Scholar
[30]Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116 (1982), 503–540.CrossRefGoogle Scholar
[31]Mañé, R.. A proof of the C 1 stability conjecture. Publ. Math. Inst. Hautes Études Sci. 66 (1988), 161–210.CrossRefGoogle Scholar
[32]Newhouse, S.. Non-density of Axiom A(a) on S2. Proc. AMS Symp. Pure Math. 14 (1970), 191–202; 335–347.CrossRefGoogle Scholar
[33]Newhouse, S.. Diffeomorphisms with infinitely many sinks. Topology 13 (1974), 9–18.CrossRefGoogle Scholar
[34]Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Asteŕisque 261 (2000), 335–347.Google Scholar
[35]Palis, J. and Smale, S.. Structural stability theorems. Global Analysis (Proceedings of Symposia on Pure Mathematics (Berkeley 1968), XIV). American Mathematical Society, Providence, RI, 1970, pp. 223–232.CrossRefGoogle Scholar
[36]Palis, J. and Viana, M.. High dimension diffeomorphisms displaying infinitely many sinks. Ann. of Math. (2) 140 (1994), 1–71.CrossRefGoogle Scholar
[37]Pliss, V.. On a conjecture due to Smale. Differ. Uravnen. 8 (1972), 268–282.Google Scholar
[38]Pujals, E. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151 (2000), 961–1023.CrossRefGoogle Scholar
[39]Pujals, E. and Sambarino, M.. Density of hyperbolicity and tangencies in sectional dissipative regions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009), 1971–2000.Google Scholar
[40]Romero, N.. Persistence of homoclinic tangencies in higher dimensions. Ergod. Th. & Dynam. Sys. 15 (1995), 735–757.CrossRefGoogle Scholar
[41]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[42]Wen, L.. On the C 1-stability conjecture for flows. J. Differential Equations 129 (1995), 334–357.CrossRefGoogle Scholar
[43]Wen, L.. Homoclinic tangencies and dominated splittings. Nonlinearity 15 (2002), 1445–1469.CrossRefGoogle Scholar
[44]Wen, L.. Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles. Bull. Braz. Math. Soc. (N.S.) 35 (2004), 419–452.CrossRefGoogle Scholar
[45]Wen, L.. The selecting lemma of Liao. Discrete Contin. Dyn. Syst. 20 (2008), 159–175.CrossRefGoogle Scholar
[46]Xia, Z. and Wen, L.. C 1 connecting lemmas. Trans. Amer. Math. Soc. 352 (2000), 5213–5230.Google Scholar
[47]Yang, D., Gan, S. and Wen, L.. Minimal non-hyperbolicity and index-completeness. Discrete Contin. Dyn. Syst. 25 (2009), 1349–1366.CrossRefGoogle Scholar
[48]Yang, J.. C 1 dynamics far from tangencies. PhD Thesis, www.preprint.impa.br.Google Scholar
[49]Yang, J.. Ergodic measure far away from tangencies. Preprint, 2007.Google Scholar