American Institute of Mathematical Sciences

Advanced
January  2016, 36(1): 469-479. doi: 10.3934/dcds.2016.36.469

Center specification property and entropy for partially hyperbolic diffeomorphisms

Lin Wang 1, and  Yujun Zhu 1,

1. 

College of Mathematics and Information Science, and Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, 050024, China

Received  September 2014 Revised  April 2015 Published  June 2015

Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $\mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, \mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $\Lambda$ is center topologically mixing then $f|_{\Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $ h(f)\leq h(f,\mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $\mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.
Keywords: center specification property, Partial hyperbolicity, uniformly compact center foliation, topological entropy..
Mathematics Subject Classification: Primary: 37D30, 37B40; Secondary: 37C5.
Citation: Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
References:
[1]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.  Google Scholar

[2]

D. Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, Journal of Modern Dynamics, 7 (2013), 565-604. doi: 10.3934/jmd.2013.7.565.  Google Scholar

[3]

C. Bonatti and D. Bohnet, Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar

[4]

C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia Math. Sci., 102, Springer, Berlin, 2005.  Google Scholar

[5]

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

[6]

M. Brin and J. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. doi: 10.1070/IM1974v008n01ABEH002101.  Google Scholar

[7]

P. D. Carrasco, Compact Dynamical Foliations, Ph.D thesis, University of Toronto, 2011.  Google Scholar

[8]

P. D. Carrasco, Compact dynamical foliations,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar

[9]

F. Hertz, J. Hertz and R. Ures, Partially Hyperbolic Dynamics, $28^0$ Coloquio Brasileiro de Matematica, IMPA Mathematical Publications, IMPA-Rio de Janeiro, 2011.  Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math. 583, Springer, New York, 1977.  Google Scholar

[11]

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.  Google Scholar

[12]

H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing and quasi-stability for dynamically coherent partially hyperbolic diffeomorphisms,, , ().   Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[14]

S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing, Discrete Continuous Dynam. Systems, 33 (2013), 2901-2909. doi: 10.3934/dcds.2013.33.2901.  Google Scholar

[15]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics. European Math. Soc., Zurich, 2004. doi: 10.4171/003.  Google Scholar

[16]

C. Pugh, M. Shub and A. Wilkinson, Holder foliations, revisited, J. Modern Dyn., 6 (2012), 79-120. doi: 10.3934/jmd.2012.6.79.  Google Scholar

[17]

P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.  Google Scholar

show all references

References:
[1]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.  Google Scholar

[2]

D. Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, Journal of Modern Dynamics, 7 (2013), 565-604. doi: 10.3934/jmd.2013.7.565.  Google Scholar

[3]

C. Bonatti and D. Bohnet, Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar

[4]

C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia Math. Sci., 102, Springer, Berlin, 2005.  Google Scholar

[5]

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

[6]

M. Brin and J. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. doi: 10.1070/IM1974v008n01ABEH002101.  Google Scholar

[7]

P. D. Carrasco, Compact Dynamical Foliations, Ph.D thesis, University of Toronto, 2011.  Google Scholar

[8]

P. D. Carrasco, Compact dynamical foliations,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar

[9]

F. Hertz, J. Hertz and R. Ures, Partially Hyperbolic Dynamics, $28^0$ Coloquio Brasileiro de Matematica, IMPA Mathematical Publications, IMPA-Rio de Janeiro, 2011.  Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math. 583, Springer, New York, 1977.  Google Scholar

[11]

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.  Google Scholar

[12]

H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing and quasi-stability for dynamically coherent partially hyperbolic diffeomorphisms,, , ().   Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[14]

S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing, Discrete Continuous Dynam. Systems, 33 (2013), 2901-2909. doi: 10.3934/dcds.2013.33.2901.  Google Scholar

[15]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics. European Math. Soc., Zurich, 2004. doi: 10.4171/003.  Google Scholar

[16]

C. Pugh, M. Shub and A. Wilkinson, Holder foliations, revisited, J. Modern Dyn., 6 (2012), 79-120. doi: 10.3934/jmd.2012.6.79.  Google Scholar

[17]

P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.  Google Scholar

[1]

Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565
[2]

José Santana Campos Costa, Fernando Micena. Pathological center foliation with dimension greater than one. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1049-1070. doi: 10.3934/dcds.2019044
[3]

Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55
[4]

Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747
[5]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[6]

Guowei Yu. Periodic solutions of the planar N-center problem with topological constraints. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5131-5162. doi: 10.3934/dcds.2016023
[7]

Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2991-3009. doi: 10.3934/dcds.2013.33.2991
[8]

Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann. On the torsion in the center conjecture. Electronic Research Announcements, 2018, 25: 27-35. doi: 10.3934/era.2018.25.004
[9]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[10]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89
[11]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
[12]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[13]

Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501
[14]

B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609
[15]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[16]

Claudio Buzzi, Claudio Pessoa, Joan Torregrosa. Piecewise linear perturbations of a linear center. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3915-3936. doi: 10.3934/dcds.2013.33.3915
[17]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51
[18]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213
[19]

Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905
[20]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences