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

Katsutoshi Shinohara

doi:10.3934/dcds.2011.31.913

Article Contents

2011, Volume 31, Issue 3: 913-940. Doi: 10.3934/dcds.2011.31.913

This issue Previous Article Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders Next Article A spectral gap for transfer operators of piecewise expanding maps

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

Katsutoshi Shinohara^1,

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

Received: January 2010

Revised: July 2011

Published: September 2011

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 37D25, 3730, Secondary: 34D09.

Citation:

Related Papers

Cited by

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, arXiv:1104.0980.
[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, arXiv:0912.1121.
[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," 2^nd 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.