# American Institute of Mathematical Sciences

September  2014, 34(9): 3789-3801. doi: 10.3934/dcds.2014.34.3789

## Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms

 1 Instituto de Matemáticas, Pontifícia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile

Received  April 2013 Revised  December 2013 Published  March 2014

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and with the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.
Citation: Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789
##### References:
 [1] F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60. doi: 10.1007/s11856-011-0041-5.  Google Scholar [2] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.  Google Scholar [3] N. Gourmelon, Adapted metrics for dominated splittings, Erg. Th. Dyn. Syst., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.  Google Scholar [4] Y. Hua, R. Saghin and Z. Xia, Topological entropy and partially hyperbolic diffeomorphisms, Erg. Th. Dynam. Syst., 28 (2008), 843-862. doi: 10.1017/S0143385707000405.  Google Scholar [5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. With a Supplementary Chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.  Google Scholar [6] O. Kozlovski, An integral formula for topological entropy of $C^{\infty}$ maps, Erg. Th. Dyn. Sys., 18 (1998), 405-424. doi: 10.1017/S0143385798100391.  Google Scholar [7] S. Newhouse, Entropy and volume, Erg. Th. Dyn. Sys., 8 (1988), 283-299. doi: 10.1017/S0143385700009469.  Google Scholar [8] F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Inv. Math., 59 (1980), 205-213. doi: 10.1007/BF01453234.  Google Scholar [9] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.  Google Scholar [10] R. Saghin, Note on homology of expanding foliations, Discr. Cont. Dyn. Sys.-Series S, 2 (2009), 349-360. doi: 10.3934/dcdss.2009.2.349.  Google Scholar [11] R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1D center, Topol. and Appl., 157 (2010), 29-34. doi: 10.1016/j.topol.2009.04.053.  Google Scholar [12] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.  Google Scholar

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##### References:
 [1] F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60. doi: 10.1007/s11856-011-0041-5.  Google Scholar [2] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.  Google Scholar [3] N. Gourmelon, Adapted metrics for dominated splittings, Erg. Th. Dyn. Syst., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.  Google Scholar [4] Y. Hua, R. Saghin and Z. Xia, Topological entropy and partially hyperbolic diffeomorphisms, Erg. Th. Dynam. Syst., 28 (2008), 843-862. doi: 10.1017/S0143385707000405.  Google Scholar [5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. With a Supplementary Chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.  Google Scholar [6] O. Kozlovski, An integral formula for topological entropy of $C^{\infty}$ maps, Erg. Th. Dyn. Sys., 18 (1998), 405-424. doi: 10.1017/S0143385798100391.  Google Scholar [7] S. Newhouse, Entropy and volume, Erg. Th. Dyn. Sys., 8 (1988), 283-299. doi: 10.1017/S0143385700009469.  Google Scholar [8] F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Inv. Math., 59 (1980), 205-213. doi: 10.1007/BF01453234.  Google Scholar [9] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.  Google Scholar [10] R. Saghin, Note on homology of expanding foliations, Discr. Cont. Dyn. Sys.-Series S, 2 (2009), 349-360. doi: 10.3934/dcdss.2009.2.349.  Google Scholar [11] R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1D center, Topol. and Appl., 157 (2010), 29-34. doi: 10.1016/j.topol.2009.04.053.  Google Scholar [12] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.  Google Scholar
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