Center specification property and entropy for partially hyperbolic diffeomorphisms

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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

Abstract
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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 and 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.
[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.
[3]	C. Bonatti and D. Bohnet, Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics, to appear in Ergodic Theory Dynam. Systems, arXiv:1210.2835.
[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.
[5]	R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math., 154 (1971), 377-397.
[6]	M. Brin and J. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. doi: 10.1070/IM1974v008n01ABEH002101.
[7]	P. D. Carrasco, Compact Dynamical Foliations, Ph.D thesis, University of Toronto, 2011.
[8]	P. D. Carrasco, Compact dynamical foliations, to appear in Ergodic Theory Dynam. Systems, arXiv:1105.0052.
[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.
[10]	M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math. 583, Springer, New York, 1977.
[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.
[12]	H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing and quasi-stability for dynamically coherent partially hyperbolic diffeomorphisms, arXiv:1405.0081.
[13]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[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.
[15]	Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics. European Math. Soc., Zurich, 2004. doi: 10.4171/003.
[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.
[17]	P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.

show all references

References:

[1]	L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.
[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.
[3]	C. Bonatti and D. Bohnet, Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics, to appear in Ergodic Theory Dynam. Systems, arXiv:1210.2835.
[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.
[5]	R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math., 154 (1971), 377-397.
[6]	M. Brin and J. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. doi: 10.1070/IM1974v008n01ABEH002101.
[7]	P. D. Carrasco, Compact Dynamical Foliations, Ph.D thesis, University of Toronto, 2011.
[8]	P. D. Carrasco, Compact dynamical foliations, to appear in Ergodic Theory Dynam. Systems, arXiv:1105.0052.
[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.
[10]	M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math. 583, Springer, New York, 1977.
[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.
[12]	H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing and quasi-stability for dynamically coherent partially hyperbolic diffeomorphisms, arXiv:1405.0081.
[13]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[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.
[15]	Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics. European Math. Soc., Zurich, 2004. doi: 10.4171/003.
[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.
[17]	P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.

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