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July  2016, 36(7): 3911-3925. doi: 10.3934/dcds.2016.36.3911

Hyperbolic periodic points for chain hyperbolic homoclinic classes

Wenxiang Sun 1, and  Yun Yang 2,

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871
2. 

School of Mathematical Sciences, Peking University, Beijing, 100871

Received  May 2015 Revised  January 2016 Published  March 2016

In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we prove that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy. We also obtain that the hyperbolic periodic measures are dense in the space of invariant measures.
Keywords: topological entropy, growth of periodic points, Chain hyperbolicity, maximal entropy measures., closing property.
Mathematics Subject Classification: 37D30, 37D20, 37C2.
Citation: Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
References:
[1]

R. Bowen, Periodic points and measures for Axiom a diffeomorphisms, Trans. Amer. Math. Soci., 154 (1971), 377-397.

[2]

Y. M. Chung and M. Hirayama, Topological entropy and periodic orbits of saddle type for surface diffeomorphisms, Hiroshima Math. J., 33 (2003), 189-195.

[3]

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

[4]

S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: A dichotomy phenomenon/mechanism for diffeomorphisms, Invent. Math., 201 (2015), 385-517. doi: 10.1007/s00222-014-0553-9.

[5]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, springer Verlag, Berlin, 1977.

[6]

H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing for partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 35 (2015), 412-430. doi: 10.1017/etds.2014.126.

[7]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Pub. Math. de l'Institut des Hautes Études Scientifiques, 51 (1980), 137-173.

[8]

S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing, Discrete Contin. Dyn. Sys., 33 (2013), 2901-2909. doi: 10.3934/dcds.2013.33.2901.

[9]

G. Liao, M. Viana and J. Yang, The Entropy Conjecture for Diffeomorphisms away from Tangencies, J. Eur. Math. Soc., 15 (2013), 2043-2060. doi: 10.4171/JEMS/413.

[10]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie Complexe et Systèmes Dynamiques, Orsay, France, 261 (2000), 335-347.

[11]

J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001.

[12]

V. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268-282.

[13]

K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.

show all references

References:
[1]

R. Bowen, Periodic points and measures for Axiom a diffeomorphisms, Trans. Amer. Math. Soci., 154 (1971), 377-397.

[2]

Y. M. Chung and M. Hirayama, Topological entropy and periodic orbits of saddle type for surface diffeomorphisms, Hiroshima Math. J., 33 (2003), 189-195.

[3]

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

[4]

S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: A dichotomy phenomenon/mechanism for diffeomorphisms, Invent. Math., 201 (2015), 385-517. doi: 10.1007/s00222-014-0553-9.

[5]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, springer Verlag, Berlin, 1977.

[6]

H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing for partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 35 (2015), 412-430. doi: 10.1017/etds.2014.126.

[7]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Pub. Math. de l'Institut des Hautes Études Scientifiques, 51 (1980), 137-173.

[8]

S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing, Discrete Contin. Dyn. Sys., 33 (2013), 2901-2909. doi: 10.3934/dcds.2013.33.2901.

[9]

G. Liao, M. Viana and J. Yang, The Entropy Conjecture for Diffeomorphisms away from Tangencies, J. Eur. Math. Soc., 15 (2013), 2043-2060. doi: 10.4171/JEMS/413.

[10]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie Complexe et Systèmes Dynamiques, Orsay, France, 261 (2000), 335-347.

[11]

J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001.

[12]

V. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268-282.

[13]

K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.

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