• Previous Article
    On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions
  • DCDS Home
  • This Issue
  • Next Article
    Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces
September  2016, 36(9): 4739-4759. doi: 10.3934/dcds.2016006

Dominated splitting, partial hyperbolicity and positive entropy

1. 

Instituto de Matemática y Estadística Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República

2. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  July 2015 Revised  March 2016 Published  May 2016

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary. We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The conditions deal with the dynamical behaviour of the (non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue measure is $\delta$-recurrent then the entropy of $f$ is positive. We give counterexamples showing that these sufficient conditions are not necessary. Finally, in the case of partially hyperbolic diffeomorphisms, we give a positive lower bound for the entropy relating it with the dimension of the unstable and stable sub-bundles.
Citation: Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
References:
[1]

L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[2]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems, Ann. of Math., 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[3]

C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III. Springer-Verlag, Berlin, 2005.

[4]

R. Bowen, Periodic point and measures for axiom-A-diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.

[5]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[6]

E. Catsigeras, M. Cerminara and H. Enrich, Pesin's entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory and Dynamical Systems, 35 (2015), 737-761. doi: 10.1017/etds.2013.93.

[7]

E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics, Bull. Pol. Acad. Sci. Math., 59 (2011), 151-164. doi: 10.4064/ba59-2-5.

[8]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 527.

[9]

L. J. Díaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 32 (2012), 4195-4207. doi: 10.3934/dcds.2012.32.4195.

[10]

N. Gourmelon, Addapted metrics for dominated splitting, Ergod. Th. and Dyn. Sys., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.

[11]

N. Gourmelon and R. Potrie, Projectively Anosov Diffeomorphisms of Surfaces, Preprint to appear. Personal communication, 2015.

[12]

G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, Journal of the European Mathematical Society, 15 (2013), 2043-2060. doi: 10.4171/JEMS/413.

[13]

V. I. Oseledec, Multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210; translated from Russian.

[14]

M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317. doi: 10.5209/rev_REMA.2008.v21.n2.16370.

[15]

H. Qiu, Existence and uniqueness of SRB measure on $C^1$ generic hyperbolic attractors, Commun. Math. Phys., 302 (2011), 345-357. doi: 10.1007/s00220-010-1160-2.

[16]

R. Saghin, W. Sun and E. Vargas, Ergodic properties of time-changes for flows, preprint.

[17]

N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proc. Amer. Math. Soc., 144 (2016), 1161-1170. doi: 10.1090/proc/12830.

[18]

W. Sun and X. Tian, Dominated splitting and Pesin's entropy formula, Discrete and Continuous Dynamical Systems, 32 (2012), 1421-1434.

[19]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.

[20]

J. Yang, $C^1$ dynamics far from tangencies, preprint.

show all references

References:
[1]

L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[2]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems, Ann. of Math., 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[3]

C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III. Springer-Verlag, Berlin, 2005.

[4]

R. Bowen, Periodic point and measures for axiom-A-diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.

[5]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[6]

E. Catsigeras, M. Cerminara and H. Enrich, Pesin's entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory and Dynamical Systems, 35 (2015), 737-761. doi: 10.1017/etds.2013.93.

[7]

E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics, Bull. Pol. Acad. Sci. Math., 59 (2011), 151-164. doi: 10.4064/ba59-2-5.

[8]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 527.

[9]

L. J. Díaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 32 (2012), 4195-4207. doi: 10.3934/dcds.2012.32.4195.

[10]

N. Gourmelon, Addapted metrics for dominated splitting, Ergod. Th. and Dyn. Sys., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.

[11]

N. Gourmelon and R. Potrie, Projectively Anosov Diffeomorphisms of Surfaces, Preprint to appear. Personal communication, 2015.

[12]

G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, Journal of the European Mathematical Society, 15 (2013), 2043-2060. doi: 10.4171/JEMS/413.

[13]

V. I. Oseledec, Multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210; translated from Russian.

[14]

M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317. doi: 10.5209/rev_REMA.2008.v21.n2.16370.

[15]

H. Qiu, Existence and uniqueness of SRB measure on $C^1$ generic hyperbolic attractors, Commun. Math. Phys., 302 (2011), 345-357. doi: 10.1007/s00220-010-1160-2.

[16]

R. Saghin, W. Sun and E. Vargas, Ergodic properties of time-changes for flows, preprint.

[17]

N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proc. Amer. Math. Soc., 144 (2016), 1161-1170. doi: 10.1090/proc/12830.

[18]

W. Sun and X. Tian, Dominated splitting and Pesin's entropy formula, Discrete and Continuous Dynamical Systems, 32 (2012), 1421-1434.

[19]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.

[20]

J. Yang, $C^1$ dynamics far from tangencies, preprint.

[1]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215

[2]

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

[3]

Zeya Mi. SRB measures for some diffeomorphisms with dominated splittings as zero noise limits. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6441-6465. doi: 10.3934/dcds.2019279

[4]

Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003

[5]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421

[6]

Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219

[7]

Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17

[8]

Dominic Veconi. SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3415-3430. doi: 10.3934/dcds.2022020

[9]

Ming-Chia Li, Ming-Jiea Lyu. Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 243-252. doi: 10.3934/dcds.2011.30.243

[10]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205

[11]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087

[12]

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

[13]

Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177

[14]

Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497

[15]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225

[16]

Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164

[17]

Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433

[18]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

[19]

Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (187)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]