American Institute of Mathematical Sciences

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. Google Scholar [2] C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, 102, (2005). Google Scholar [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. Google Scholar [4] J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity,, J. Mod. Dyn., 7 (2013), 527. doi: 10.3934/jmd.2013.7.527. Google Scholar [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. Google Scholar [6] K. Gelfert, Somersaults on unstable islands, preprint,, , (). Google Scholar [7] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977). Google Scholar [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. Google Scholar [9] A. Katok, A conjecture about entropy,, AMS. Transl, 133 (1986), 91. Google Scholar [10] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [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. Google Scholar [12] R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. Google Scholar [13] R. Mañé, IntroduÇão à Teoria Ergódica,, Projeto Euclides [Euclid Project], (1983). Google Scholar [14] M. Misiurewicz and F. Przytycki, Entropy conjecture for tori,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25 (1977), 575. Google Scholar [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. Google Scholar [16] D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6. Google Scholar [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. Google Scholar [18] M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. Google Scholar [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. Google Scholar [20] P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0. Google Scholar [21] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar [22] Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215. Google Scholar

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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. Google Scholar [2] C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, 102, (2005). Google Scholar [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. Google Scholar [4] J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity,, J. Mod. Dyn., 7 (2013), 527. doi: 10.3934/jmd.2013.7.527. Google Scholar [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. Google Scholar [6] K. Gelfert, Somersaults on unstable islands, preprint,, , (). Google Scholar [7] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977). Google Scholar [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. Google Scholar [9] A. Katok, A conjecture about entropy,, AMS. Transl, 133 (1986), 91. Google Scholar [10] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [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. Google Scholar [12] R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. Google Scholar [13] R. Mañé, IntroduÇão à Teoria Ergódica,, Projeto Euclides [Euclid Project], (1983). Google Scholar [14] M. Misiurewicz and F. Przytycki, Entropy conjecture for tori,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25 (1977), 575. Google Scholar [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. Google Scholar [16] D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6. Google Scholar [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. Google Scholar [18] M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. Google Scholar [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. Google Scholar [20] P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0. Google Scholar [21] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar [22] Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215. Google Scholar
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