Dominated splitting, partial hyperbolicity and positive entropy

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September 2016, 36(9): 4739-4759. doi: 10.3934/dcds.2016006

Dominated splitting, partial hyperbolicity and positive entropy

Eleonora Catsigeras ^1, and Xueting Tian ^2,

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

Abstract
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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.

Keywords: SRB-like measures., SRB, volume, smooth, positive topological entropy and positive metric entropy, Dominated splitting and partial hyperbolicity.

Mathematics Subject Classification: Primary: 37D30, 37B40, 37D25; Secondary: 37A3.

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.

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