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C1-stably shadowable chain components

Published online by Cambridge University Press:  01 June 2008

KAZUHIRO SAKAI
KAZUHIRO SAKAI*
Affiliation:
Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505, Japan (email: kazsakai@cc.utsunomiya-u.ac.jp)
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Abstract

Let p be a hyperbolic periodic saddle of a diffeomorphism f on a closed manifold M, and let Cf(p) be the chain component of f containing p. In this paper, we show that if Cf(p) is C1-stably shadowable, then (i) Cf(p) is the homoclinic class of p and admits a dominated splitting (where the dimension of E is equal to that of the stable eigenspace of p); (ii) the Cf(p)-germ of f is expansive if and only if Cf(p) is hyperbolic; and (iii) when M is a surface, Cf(p) is locally maximal if and only if Cf (p)is hyperbolic.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 987 - 1029
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Abdenur, F.. Generic robustness of spectral decompositions. Ann. Sc. École Norm. Sup., Ser. 4 36 (2003), 213–224.CrossRefGoogle Scholar
[2]Abdenur, F., Bonatti, C., Crovisier, S. and Díaz, L. J.. Generic diffeomorphisms on compact surfaces. Fund. Math. 187 (2005), 127–159.CrossRefGoogle Scholar
[3]Abdenur, F. and Díaz, L. J.. Pseudo-orbit shadowing in the C 1 topology. Discrete Contin. Dyn. Syst. 17 (2007), 223–245.CrossRefGoogle Scholar
[4]Aoki, N.. On homeomorphisms with pseudo-orbit tracing property. Tokyo J. Math. 6 (1983), 329–334.CrossRefGoogle Scholar
[5]Bonatti, C. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158 (2004), 33–104.CrossRefGoogle Scholar
[6]Bonatti, C. and Díaz, L. J.. Persistence of transitive diffeomorphisms. Ann. of Math. 143 (1995), 367–396.Google Scholar
[7]Bonatti, C., Díaz, L. J. and Turcat, G.. Pas de ≪Shadowing lemma≫ pour les dynamiques partiellment hyperboliques. C. R. Acad. Sci. Paris, Ser. I 330 (2000), 587–592.CrossRefGoogle Scholar
[8]Bonatti, C., Díaz, L. J. and Pujals, E. R.. A C 1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. 158 (2003), 355–418.CrossRefGoogle Scholar
[9]Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopedia of Mathematical Sciences, Mathematical Physics, 102). Springer, Berlin, 2005.Google Scholar
[10]Burns, K., Pugh, C., Shub, M. and Wilkinson, A.. Recent results about stable ergodicity. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings Symposia in Pure Mathematics, 69). American Mathematical Society, Providence, RI, 2001, pp. 327–366.CrossRefGoogle Scholar
[11]Conley, C.. Isolated Invariant Sets and Morse Index (CBMS Regional Conferences Series in Mathematics, 38). American Mathematical Society, Providence, RI, 1978.CrossRefGoogle Scholar
[12]Crovisier, S.. Saddle-node bifurcations for hyperbolic sets. Ergod. Th. & Dynam. Sys. 22 (2002), 1079–1115.CrossRefGoogle Scholar
[13]Díaz, L. J.. Robust nonhyperbolic dynamics and heterodimensional cycles. Ergod. Th. & Dynam. Sys. 15 (1995), 291–315.CrossRefGoogle Scholar
[14]Díaz, L. J., Pujals, E. R. and Ures, R.. Partial hyperbolicity and robust transitivity. Acta Math. 183 (1999), 1–43.CrossRefGoogle Scholar
[15]Díaz, L. J. and Rocha, J.. Partially hyperbolic and transitive dynamics generated by heteroclinic cycles. Ergod. Th. & Dynam. Sys. 21 (2001), 25–76.CrossRefGoogle Scholar
[16]Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301–308.CrossRefGoogle Scholar
[17]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
[18]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[19]Mañé, R.. Expansive diffeomorphisms. Proc. Symp. on Dynamical Systems (University of Warwick, 1974) (Lecture Notes in Mathematics, 468). 1975, pp. 162–174.Google Scholar
[20]Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.CrossRefGoogle Scholar
[21]Mañé, R.. Persistent manifolds are normally hyperbolic. Trans. Amer. Math. Soc. 246 (1978), 261–283.CrossRefGoogle Scholar
[22]Mañé, R.. An ergodic closing lemma. Ann. of Math. 116 (1982), 503–540.CrossRefGoogle Scholar
[23]Moriyasu, K.. The topological stability of diffeomorphisms. Nagoya Math. J. 123 (1991), 91–102.CrossRefGoogle Scholar
[24]Pacifico, M. J., Pujals, E. R. and Vieitez, J. L.. Robustly expansive homoclinic classes. Ergod. Th. & Dynam. Sys. 25 (2005), 271–300.CrossRefGoogle Scholar
[25]Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge Studies in Advanced Mathematics, 35). Cambridge University Press, Cambridge, 1993.Google Scholar
[26]Pilyugin, S. Yu.. Shadowing in Dynamical Systems (Lecture Notes in Mathmatics, 1706). Springer, Berlin, 1999.Google Scholar
[27]Pilyugin, S. Yu., Rodionova, A. A. and Sakai, K.. Orbital and weak shadowing properties. Discrete Contin. Dyn. Syst. 9 (2003), 287–308.CrossRefGoogle Scholar
[28]Pilyugin, S. Yu. and Sakai, K.. C 0 transversality and shadowing properties. Proc. Steklov Inst. Math. 256 (2007), 290–305.CrossRefGoogle Scholar
[29]Pujals, E. R. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. 151 (2000), 961–1023.CrossRefGoogle Scholar
[30]Robinson, C.. Stability theorems and hyperbolicity in dynamical systems. Rocky Mountain J. Math. 7 (1977), 425–437.CrossRefGoogle Scholar
[31]Sakai, K.. Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds. Osaka J. Math. 31 (1994), 373–386.Google Scholar
[32]Sakai, K.. Shadowing Property and Transversality Condition (Proc. Int. Conf. on Dynamical Systems and Chaos, 1). World Scientific, Singapore, 1995, pp. 233–238.Google Scholar
[33]Sakai, K.. Diffeomorphisms with C 2 stable shadowing. Dyn. Syst. 17 (2002), 235–241.CrossRefGoogle Scholar
[34]Sambarino, M. and Vieitez, J.. On C 1-persistently expansive homoclinic classes. Discrete Contin. Dyn. Syst. 14 (2006), 465–481.CrossRefGoogle Scholar
[35]Shimomura, T.. On a structure of discrete dynamical systems from the view point of chain components and some applications. Japan. J. Math. (N. S.) 15 (1989), 99–126.CrossRefGoogle Scholar
[36]Smale, S.. Diffeomorphisms with many periodic points. Differential and Combinatorial Topology. Princeton University Press, Princeton, NJ, 1965, pp. 63–80.CrossRefGoogle Scholar
[37]Wen, L.. Homoclinic tangencies and dominated splittings. Nonlinearity 15 (2002), 1445–1469.CrossRefGoogle Scholar
[38]Yuan, G.-C. and Yorke, J. A.. An open set of maps for which every point is absolutely nonshadowable. Proc. Amer. Math. Soc. 128 (2000), 909–918.CrossRefGoogle Scholar