Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-04T19:01:09.941Z Has data issue: false hasContentIssue false
  • English
  • Français

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: yangjg@impa.br)
Get access Rights & Permissions [Opens in a new window]

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

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

[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