September  2011, 31(3): 913-940. doi: 10.3934/dcds.2011.31.913

On the index problem of $C^1$-generic wild homoclinic classes in dimension three

1. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1- Komaba Meguro-ku Tokyo 153-8914, Japan

Received  January 2010 Revised  July 2011 Published  August 2011

We study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that, $C^1$-generically, if such a homoclinic class contains a volume-expanding periodic point, then it contains a hyperbolic periodic point whose index (dimension of the unstable manifold) is equal to two.
Citation: Katsutoshi Shinohara. On the index problem of $C^1$-generic wild homoclinic classes in dimension three. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 913-940. doi: 10.3934/dcds.2011.31.913
References:
[1]

F. Abdenur, Ch. Bonatti, S. Crovisier, L. Díaz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory Dynam. Systems, 27 (2007), 1.  doi: 10.1017/S0143385706000538.  Google Scholar

[2]

C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33.   Google Scholar

[3]

C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171.   Google Scholar

[4]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469.  doi: 10.1017/S1474748008000030.  Google Scholar

[5]

C. Bonatti, L. Díaz and S. Kiriki, Stabilization of heterodimensional cycles,, preprint, ().   Google Scholar

[6]

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. (2), 158 (2003), 355.  doi: 10.4007/annals.2003.158.355.  Google Scholar

[7]

C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,", Encyclopaedia of Mathematical Sciences, 102 (2005).   Google Scholar

[8]

S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index,, Trends in Mathematics, 7 (2004), 143.   Google Scholar

[9]

N. Gourmelon, A Franks' lemma that preserves invariant manifolds,, Preprint, ().   Google Scholar

[10]

N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations,, Discrete Contin. Dyn. Syst., 26 (2010), 1.  doi: 10.3934/dcds.2010.26.1.  Google Scholar

[11]

R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503.   Google Scholar

[12]

J. Palis, Open questions leading to a global perspective in dynamics,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/4/T01.  Google Scholar

[13]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Ann. of Math. (2), 151 (2000), 961.  doi: 10.2307/121127.  Google Scholar

[14]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,", 2nd edition, (1999).   Google Scholar

[15]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[16]

K. Shinohara, An example of C1-generically wild homoclinic classes with index deficiency,, Nonlinearity, 24 (2011), 1961.  doi: 10.1088/0951-7715/24/7/003.  Google Scholar

[17]

L. Wen, Homoclinic tangencies and dominated splittings,, Nonlinearity, 15 (2002), 1445.  doi: 10.1088/0951-7715/15/5/306.  Google Scholar

show all references

References:
[1]

F. Abdenur, Ch. Bonatti, S. Crovisier, L. Díaz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory Dynam. Systems, 27 (2007), 1.  doi: 10.1017/S0143385706000538.  Google Scholar

[2]

C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33.   Google Scholar

[3]

C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171.   Google Scholar

[4]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469.  doi: 10.1017/S1474748008000030.  Google Scholar

[5]

C. Bonatti, L. Díaz and S. Kiriki, Stabilization of heterodimensional cycles,, preprint, ().   Google Scholar

[6]

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. (2), 158 (2003), 355.  doi: 10.4007/annals.2003.158.355.  Google Scholar

[7]

C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,", Encyclopaedia of Mathematical Sciences, 102 (2005).   Google Scholar

[8]

S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index,, Trends in Mathematics, 7 (2004), 143.   Google Scholar

[9]

N. Gourmelon, A Franks' lemma that preserves invariant manifolds,, Preprint, ().   Google Scholar

[10]

N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations,, Discrete Contin. Dyn. Syst., 26 (2010), 1.  doi: 10.3934/dcds.2010.26.1.  Google Scholar

[11]

R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503.   Google Scholar

[12]

J. Palis, Open questions leading to a global perspective in dynamics,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/4/T01.  Google Scholar

[13]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Ann. of Math. (2), 151 (2000), 961.  doi: 10.2307/121127.  Google Scholar

[14]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,", 2nd edition, (1999).   Google Scholar

[15]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[16]

K. Shinohara, An example of C1-generically wild homoclinic classes with index deficiency,, Nonlinearity, 24 (2011), 1961.  doi: 10.1088/0951-7715/24/7/003.  Google Scholar

[17]

L. Wen, Homoclinic tangencies and dominated splittings,, Nonlinearity, 15 (2002), 1445.  doi: 10.1088/0951-7715/15/5/306.  Google Scholar

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