March  2016, 36(3): 1693-1707. doi: 10.3934/dcds.2016.36.1693

Structurally stable homoclinic classes

1. 

School of Mathematics and Systems Science, Beihang University, Beijing 100191

Received  October 2014 Revised  June 2015 Published  August 2015

In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the classes is not innately locally maximal, it is hard to answer whether structurally stable homoclinic classes are hyperbolic. In this article, we make some progress on this question. We prove that if a homoclinic class is structurally stable, then it admits a dominated splitting. Moreover we prove that codimension one structurally stable classes are hyperbolic. Also, if the diffeomorphism is far away from homoclinic tangencies, then structurally stable homoclinic classes are hyperbolic.
Citation: Xiao Wen. Structurally stable homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1693-1707. doi: 10.3934/dcds.2016.36.1693
References:
[1]

F. Abdenur, Ch. Bonatti, S. Crovisier, L. J. Diáz and L. Wen, Periodic points and homoclinic classes, Ergodic Theory and Dynamical Systems, 27 (2007), 1-22. doi: 10.1017/S0143385706000538.

[2]

C. Bonatti and S. Crovisier, Recurrence and genericity, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.

[3]

C. Bonatti L. Díaz, and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418. doi: 10.4007/annals.2003.158.355.

[4]

C. Bonatti, S. Gan and D. Yang, On the hyperbolicity of homoclinic classes, Discrete Contin. Dyn. Syst., 25 (2009), 1143-1162. doi: 10.3934/dcds.2009.25.1143.

[5]

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Mathematics, 226 (2011), 673-726. doi: 10.1016/j.aim.2010.07.013.

[6]

S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. I.H.E.S, 104 (2006), 87-141. doi: 10.1007/s10240-006-0002-4.

[7]

S. Crovisier, M. Sambarino and D. Yang, Partial hyperbolicity and homoclinic tangencies, Journal of the European Mathematical Society, 17 (2015), 1-49. doi: 10.4171/JEMS/497.

[8]

J. Franks, Necessary conditions for stability of diffeomorphisms, Transactions of the A.M.S 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.

[9]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math., 583, Springer-Verlag, Berlin-Newyork, 1977.

[10]

S. Gan and L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dynam. Differential Equations, 15 (2003), 451-471. doi: 10.1023/B:JODY.0000009743.10365.9d.

[11]

N. Gourmelon, An Isotopic Perturbation Lemma Along Periodic Orbits manifolds,, preprint, (). 

[12]

S. Liao, A basic property of a certain class of differential systems, Acta Math. Sinica, 22 (1979), 316-343.

[13]

S. Liao, Obstruction sets II, Acta Sci. Natur. Univ. Pekinensis, 2 (1981), 1-36.

[14]

R. Mañé, An ergodic closing lemma, Annals of Math., 116 (1982), 503-540. doi: 10.2307/2007021.

[15]

R. Mañé, A proof of the $C^1$ stability conjecture, Publ. Math. I.H.E.S, 66 (1988), 161-210.

[16]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Transl. from the Portuguese by A. K. Manning. (English), Springer-Verlag, New York, 1982.

[17]

J. Palis and S. Smale, Structural stability theorems, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, 1968), Amer. Math. Soc., Providence, RI, 1970, 223-231.

[18]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math., 151 (2000), 961-1023. doi: 10.2307/121127.

[19]

M. Sambarino and J. Vieitez, On $C^1$-persistently expansive homoclinic classes, Discrete Contin. Dyn. Syst., 14 (2006), 465-481.

[20]

X. Wen and L. Wen, Codimension One Structurally Stable Chain Classes,, to appear in Transactions of the A.M.S., ().  doi: 10.1090/tran/6440.

show all references

References:
[1]

F. Abdenur, Ch. Bonatti, S. Crovisier, L. J. Diáz and L. Wen, Periodic points and homoclinic classes, Ergodic Theory and Dynamical Systems, 27 (2007), 1-22. doi: 10.1017/S0143385706000538.

[2]

C. Bonatti and S. Crovisier, Recurrence and genericity, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.

[3]

C. Bonatti L. Díaz, and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418. doi: 10.4007/annals.2003.158.355.

[4]

C. Bonatti, S. Gan and D. Yang, On the hyperbolicity of homoclinic classes, Discrete Contin. Dyn. Syst., 25 (2009), 1143-1162. doi: 10.3934/dcds.2009.25.1143.

[5]

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Mathematics, 226 (2011), 673-726. doi: 10.1016/j.aim.2010.07.013.

[6]

S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. I.H.E.S, 104 (2006), 87-141. doi: 10.1007/s10240-006-0002-4.

[7]

S. Crovisier, M. Sambarino and D. Yang, Partial hyperbolicity and homoclinic tangencies, Journal of the European Mathematical Society, 17 (2015), 1-49. doi: 10.4171/JEMS/497.

[8]

J. Franks, Necessary conditions for stability of diffeomorphisms, Transactions of the A.M.S 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.

[9]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math., 583, Springer-Verlag, Berlin-Newyork, 1977.

[10]

S. Gan and L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dynam. Differential Equations, 15 (2003), 451-471. doi: 10.1023/B:JODY.0000009743.10365.9d.

[11]

N. Gourmelon, An Isotopic Perturbation Lemma Along Periodic Orbits manifolds,, preprint, (). 

[12]

S. Liao, A basic property of a certain class of differential systems, Acta Math. Sinica, 22 (1979), 316-343.

[13]

S. Liao, Obstruction sets II, Acta Sci. Natur. Univ. Pekinensis, 2 (1981), 1-36.

[14]

R. Mañé, An ergodic closing lemma, Annals of Math., 116 (1982), 503-540. doi: 10.2307/2007021.

[15]

R. Mañé, A proof of the $C^1$ stability conjecture, Publ. Math. I.H.E.S, 66 (1988), 161-210.

[16]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Transl. from the Portuguese by A. K. Manning. (English), Springer-Verlag, New York, 1982.

[17]

J. Palis and S. Smale, Structural stability theorems, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, 1968), Amer. Math. Soc., Providence, RI, 1970, 223-231.

[18]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math., 151 (2000), 961-1023. doi: 10.2307/121127.

[19]

M. Sambarino and J. Vieitez, On $C^1$-persistently expansive homoclinic classes, Discrete Contin. Dyn. Syst., 14 (2006), 465-481.

[20]

X. Wen and L. Wen, Codimension One Structurally Stable Chain Classes,, to appear in Transactions of the A.M.S., ().  doi: 10.1090/tran/6440.

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