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

Center specification property and entropy for partially hyperbolic diffeomorphisms

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 $h(f)= p^{c}(f)$.
Citation: Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
References:
[1]

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

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

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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

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

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

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

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P. D. Carrasco, Compact Dynamical Foliations,, Ph.D thesis, (2011).   Google Scholar

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

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F. Hertz, J. Hertz and R. Ures, Partially Hyperbolic Dynamics,, $28^0$ Coloquio Brasileiro de Matematica, (2011).   Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lect. Notes in Math. 583, (1977).   Google Scholar

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

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

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

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S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing,, Discrete Continuous Dynam. Systems, 33 (2013), 2901.  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., (2004).  doi: 10.4171/003.  Google Scholar

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

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P. Walters, An Introduction to Ergodic Theory,, Springer, (1982).   Google Scholar

show all references

References:
[1]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity,, Cambridge University Press, (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.  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., (2005).   Google Scholar

[5]

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

[6]

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

[7]

P. D. Carrasco, Compact Dynamical Foliations,, Ph.D thesis, (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, (2011).   Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lect. Notes in Math. 583, (1977).   Google Scholar

[11]

H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing for partially hyperbolic diffeomorphisms,, Ergodic Theory Dynam. Systems, 35 (2015), 412.  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, (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.  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., (2004).  doi: 10.4171/003.  Google Scholar

[16]

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

[17]

P. Walters, An Introduction to Ergodic Theory,, Springer, (1982).   Google Scholar

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