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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 & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
##### References:
 [1] L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity,, Cambridge Univ. Press, (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar [2] J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems,, Ann. of Math., 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar [3] C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005).   Google Scholar [4] R. Bowen, Periodic point and measures for axiom-A-diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.   Google Scholar [5] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar [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.  doi: 10.1017/etds.2013.93.  Google Scholar [7] E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics,, Bull. Pol. Acad. Sci. Math., 59 (2011), 151.  doi: 10.4064/ba59-2-5.  Google Scholar [8] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space,, Lecture Notes in Mathematics, ().   Google Scholar [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.  doi: 10.3934/dcds.2012.32.4195.  Google Scholar [10] N. Gourmelon, Addapted metrics for dominated splitting,, Ergod. Th. and Dyn. Sys., 27 (2007), 1839.  doi: 10.1017/S0143385707000272.  Google Scholar [11] N. Gourmelon and R. Potrie, Projectively Anosov Diffeomorphisms of Surfaces,, Preprint to appear. Personal communication, (2015).   Google Scholar [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.  doi: 10.4171/JEMS/413.  Google Scholar [13] V. I. Oseledec, Multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 179.   Google Scholar [14] M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms,, Rev. Mat. Complut., 21 (2008), 293.  doi: 10.5209/rev_REMA.2008.v21.n2.16370.  Google Scholar [15] H. Qiu, Existence and uniqueness of SRB measure on $C^1$ generic hyperbolic attractors,, Commun. Math. Phys., 302 (2011), 345.  doi: 10.1007/s00220-010-1160-2.  Google Scholar [16] R. Saghin, W. Sun and E. Vargas, Ergodic properties of time-changes for flows,, preprint., ().   Google Scholar [17] N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification,, Proc. Amer. Math. Soc., 144 (2016), 1161.  doi: 10.1090/proc/12830.  Google Scholar [18] W. Sun and X. Tian, Dominated splitting and Pesin's entropy formula,, Discrete and Continuous Dynamical Systems, 32 (2012), 1421.   Google Scholar [19] P. Walters, An Introduction to Ergodic Theory,, Springer-Verlag, (1982).   Google Scholar [20] J. Yang, $C^1$ dynamics far from tangencies,, preprint., ().   Google Scholar

show all references

##### References:
 [1] L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity,, Cambridge Univ. Press, (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar [2] J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems,, Ann. of Math., 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar [3] C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005).   Google Scholar [4] R. Bowen, Periodic point and measures for axiom-A-diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.   Google Scholar [5] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar [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.  doi: 10.1017/etds.2013.93.  Google Scholar [7] E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics,, Bull. Pol. Acad. Sci. Math., 59 (2011), 151.  doi: 10.4064/ba59-2-5.  Google Scholar [8] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space,, Lecture Notes in Mathematics, ().   Google Scholar [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.  doi: 10.3934/dcds.2012.32.4195.  Google Scholar [10] N. Gourmelon, Addapted metrics for dominated splitting,, Ergod. Th. and Dyn. Sys., 27 (2007), 1839.  doi: 10.1017/S0143385707000272.  Google Scholar [11] N. Gourmelon and R. Potrie, Projectively Anosov Diffeomorphisms of Surfaces,, Preprint to appear. Personal communication, (2015).   Google Scholar [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.  doi: 10.4171/JEMS/413.  Google Scholar [13] V. I. Oseledec, Multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 179.   Google Scholar [14] M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms,, Rev. Mat. Complut., 21 (2008), 293.  doi: 10.5209/rev_REMA.2008.v21.n2.16370.  Google Scholar [15] H. Qiu, Existence and uniqueness of SRB measure on $C^1$ generic hyperbolic attractors,, Commun. Math. Phys., 302 (2011), 345.  doi: 10.1007/s00220-010-1160-2.  Google Scholar [16] R. Saghin, W. Sun and E. Vargas, Ergodic properties of time-changes for flows,, preprint., ().   Google Scholar [17] N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification,, Proc. Amer. Math. Soc., 144 (2016), 1161.  doi: 10.1090/proc/12830.  Google Scholar [18] W. Sun and X. Tian, Dominated splitting and Pesin's entropy formula,, Discrete and Continuous Dynamical Systems, 32 (2012), 1421.   Google Scholar [19] P. Walters, An Introduction to Ergodic Theory,, Springer-Verlag, (1982).   Google Scholar [20] J. Yang, $C^1$ dynamics far from tangencies,, preprint., ().   Google Scholar
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