June  2016, 36(6): 3417-3433. doi: 10.3934/dcds.2016.36.3417

Hyperbolic sets and entropy at the homological level

1. 

Departamento de Matemática PUC-Rio, Marquês de São Vicente, 225, Rio de Janeiro, 22451-900, Brazil

Received  August 2015 Revised  October 2015 Published  December 2015

The aim of this work is to study a kind of refinement of the entropy conjecture, in the context of partially hyperbolic diffeomorphism with one dimensional central direction, of $d$-dimensional torus. We start by establishing a connection between the unstable index of hyperbolic sets and the index at algebraic level. Two examples are given which might shed light on which are the good questions in the higher dimensional center case.
Citation: Mario Roldan. Hyperbolic sets and entropy at the homological level. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3417-3433. doi: 10.3934/dcds.2016.36.3417
References:
[1]

C. Bonatti, S. Crovisier and K. Shinohara, The $C^{1+\alpha}$ hypothesis in Pesin theory revisited,, Journal of Modern Dynanics, 7 (2013), 605. doi: 10.3934/jmd.2013.7.605.

[2]

C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, 102, (2005).

[3]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. Math., 115 (2000), 157. doi: 10.1007/BF02810585.

[4]

J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity,, J. Mod. Dyn., 7 (2013), 527. doi: 10.3934/jmd.2013.7.527.

[5]

T. Fisher, R. Potrie and M. Sambarino, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov,, Mathematische Zeitschrift, 278 (2014), 149. doi: 10.1007/s00209-014-1310-x.

[6]

K. Gelfert, Somersaults on unstable islands, preprint,, , ().

[7]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977).

[8]

Y. Hua, R. Saghin and Z. Xia, Topological entropy and partially hyperbolic diffeomorphisms,, Ergodic Theory Dynam. Systems, 28 (2008), 843. doi: 10.1017/S0143385707000405.

[9]

A. Katok, A conjecture about entropy,, AMS. Transl, 133 (1986), 91.

[10]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.

[11]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). doi: 10.1017/CBO9780511809187.

[12]

R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8.

[13]

R. Mañé, IntroduÇão à Teoria Ergódica,, Projeto Euclides [Euclid Project], (1983).

[14]

M. Misiurewicz and F. Przytycki, Entropy conjecture for tori,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25 (1977), 575.

[15]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves,, Ergodic Theory Dynam. Systems, 32 (2012), 825. doi: 10.1017/S0143385711000757.

[16]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6.

[17]

R. Saghin, Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms,, Discrete Contin. Dyn. Syst., 34 (2014), 3789. doi: 10.3934/dcds.2014.34.3789.

[18]

M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6.

[19]

R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part,, Proc. Amer. Math. Soc., 140 (2012), 1973. doi: 10.1090/S0002-9939-2011-11040-2.

[20]

P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0.

[21]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).

[22]

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

show all references

References:
[1]

C. Bonatti, S. Crovisier and K. Shinohara, The $C^{1+\alpha}$ hypothesis in Pesin theory revisited,, Journal of Modern Dynanics, 7 (2013), 605. doi: 10.3934/jmd.2013.7.605.

[2]

C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, 102, (2005).

[3]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. Math., 115 (2000), 157. doi: 10.1007/BF02810585.

[4]

J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity,, J. Mod. Dyn., 7 (2013), 527. doi: 10.3934/jmd.2013.7.527.

[5]

T. Fisher, R. Potrie and M. Sambarino, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov,, Mathematische Zeitschrift, 278 (2014), 149. doi: 10.1007/s00209-014-1310-x.

[6]

K. Gelfert, Somersaults on unstable islands, preprint,, , ().

[7]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977).

[8]

Y. Hua, R. Saghin and Z. Xia, Topological entropy and partially hyperbolic diffeomorphisms,, Ergodic Theory Dynam. Systems, 28 (2008), 843. doi: 10.1017/S0143385707000405.

[9]

A. Katok, A conjecture about entropy,, AMS. Transl, 133 (1986), 91.

[10]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.

[11]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). doi: 10.1017/CBO9780511809187.

[12]

R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8.

[13]

R. Mañé, IntroduÇão à Teoria Ergódica,, Projeto Euclides [Euclid Project], (1983).

[14]

M. Misiurewicz and F. Przytycki, Entropy conjecture for tori,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25 (1977), 575.

[15]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves,, Ergodic Theory Dynam. Systems, 32 (2012), 825. doi: 10.1017/S0143385711000757.

[16]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6.

[17]

R. Saghin, Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms,, Discrete Contin. Dyn. Syst., 34 (2014), 3789. doi: 10.3934/dcds.2014.34.3789.

[18]

M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6.

[19]

R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part,, Proc. Amer. Math. Soc., 140 (2012), 1973. doi: 10.1090/S0002-9939-2011-11040-2.

[20]

P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0.

[21]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).

[22]

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

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