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 and 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.

[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.

[3]

N. Gourmelon, Adapted metrics for dominated splittings, Erg. Th. Dyn. Syst., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.

[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.

[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.

[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.

[7]

S. Newhouse, Entropy and volume, Erg. Th. Dyn. Sys., 8 (1988), 283-299. doi: 10.1017/S0143385700009469.

[8]

F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Inv. Math., 59 (1980), 205-213. doi: 10.1007/BF01453234.

[9]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[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.

[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.

[12]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

show all references

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.

[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.

[3]

N. Gourmelon, Adapted metrics for dominated splittings, Erg. Th. Dyn. Syst., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.

[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.

[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.

[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.

[7]

S. Newhouse, Entropy and volume, Erg. Th. Dyn. Sys., 8 (1988), 283-299. doi: 10.1017/S0143385700009469.

[8]

F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Inv. Math., 59 (1980), 205-213. doi: 10.1007/BF01453234.

[9]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[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.

[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.

[12]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

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