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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

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 & 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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