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 - A, 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. 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. doi: 10.4007/annals.2010.171.451.

[3]

N. Gourmelon, Adapted metrics for dominated splittings,, Erg. Th. Dyn. Syst., 27 (2007), 1839. 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. 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 (1995).

[6]

O. Kozlovski, An integral formula for topological entropy of $C^{\infty}$ maps,, Erg. Th. Dyn. Sys., 18 (1998), 405. doi: 10.1017/S0143385798100391.

[7]

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

[8]

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

[9]

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

[10]

R. Saghin, Note on homology of expanding foliations,, Discr. Cont. Dyn. Sys.-Series S, 2 (2009), 349. 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. doi: 10.1016/j.topol.2009.04.053.

[12]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. 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. 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. doi: 10.4007/annals.2010.171.451.

[3]

N. Gourmelon, Adapted metrics for dominated splittings,, Erg. Th. Dyn. Syst., 27 (2007), 1839. 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. 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 (1995).

[6]

O. Kozlovski, An integral formula for topological entropy of $C^{\infty}$ maps,, Erg. Th. Dyn. Sys., 18 (1998), 405. doi: 10.1017/S0143385798100391.

[7]

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

[8]

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

[9]

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

[10]

R. Saghin, Note on homology of expanding foliations,, Discr. Cont. Dyn. Sys.-Series S, 2 (2009), 349. 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. doi: 10.1016/j.topol.2009.04.053.

[12]

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

[1]

Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195

[2]

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

[3]

François Ledrappier, Seonhee Lim. Volume entropy of hyperbolic buildings. Journal of Modern Dynamics, 2010, 4 (1) : 139-165. doi: 10.3934/jmd.2010.4.139

[4]

Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037

[5]

Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419

[6]

Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008

[7]

Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245

[8]

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

[9]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75

[10]

Dmitri Burago, Sergei Ivanov. Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups. Journal of Modern Dynamics, 2008, 2 (4) : 541-580. doi: 10.3934/jmd.2008.2.541

[11]

Keith Burns, Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Anna Talitskaya, Raúl Ures. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 75-88. doi: 10.3934/dcds.2008.22.75

[12]

Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1

[13]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869

[14]

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

[15]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341

[16]

Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611

[17]

Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228

[18]

Paweł G. Walczak. Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4731-4742. doi: 10.3934/dcds.2013.33.4731

[19]

Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204

[20]

Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]