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 and Continuous Dynamical Systems, 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), 1-22. doi: 10.1017/S0143385706000538.

[2]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.

[3]

C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171-197.

[4]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525. doi: 10.1017/S1474748008000030.

[5]

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

[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), 355-418. doi: 10.4007/annals.2003.158.355.

[7]

C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective," Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.

[8]

S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index, Trends in Mathematics, 7 (2004), 143-168.

[9]

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

[10]

N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations, Discrete Contin. Dyn. Syst., 26, (2010), 1-42. doi: 10.3934/dcds.2010.26.1.

[11]

R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.

[12]

J. Palis, Open questions leading to a global perspective in dynamics, Nonlinearity, 21 (2008), T37-T43. doi: 10.1088/0951-7715/21/4/T01.

[13]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. (2), 151 (2000), 961-1023. doi: 10.2307/121127.

[14]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos," 2nd edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999.

[15]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.

[16]

K. Shinohara, An example of C1-generically wild homoclinic classes with index deficiency, Nonlinearity, 24 (2011), 1961-1974. doi: 10.1088/0951-7715/24/7/003.

[17]

L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306.

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), 1-22. doi: 10.1017/S0143385706000538.

[2]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.

[3]

C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171-197.

[4]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525. doi: 10.1017/S1474748008000030.

[5]

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

[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), 355-418. doi: 10.4007/annals.2003.158.355.

[7]

C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective," Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.

[8]

S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index, Trends in Mathematics, 7 (2004), 143-168.

[9]

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

[10]

N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations, Discrete Contin. Dyn. Syst., 26, (2010), 1-42. doi: 10.3934/dcds.2010.26.1.

[11]

R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.

[12]

J. Palis, Open questions leading to a global perspective in dynamics, Nonlinearity, 21 (2008), T37-T43. doi: 10.1088/0951-7715/21/4/T01.

[13]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. (2), 151 (2000), 961-1023. doi: 10.2307/121127.

[14]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos," 2nd edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999.

[15]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.

[16]

K. Shinohara, An example of C1-generically wild homoclinic classes with index deficiency, Nonlinearity, 24 (2011), 1961-1974. doi: 10.1088/0951-7715/24/7/003.

[17]

L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306.

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