# American Institue of Mathematical Sciences

2012, 32(12): 4195-4207. doi: 10.3934/dcds.2012.32.4195

## Entropy-expansiveness for partially hyperbolic diffeomorphisms

 1 Dep. Matemática PUC-Rio, Marquês de São Vicente 225 22453-900, Rio de Janeiro, Brazil 2 Department of Mathematics, Brigham Young University, Provo, UT 84602 3 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970, Rio de Janeiro, Brazil 4 Instituto de Matematica, Regional Norte, Rivera 1350, Universidad de la Republica, CP 50000, Salto, Uruguay

Received  June 2011 Revised  November 2011 Published  August 2012

We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.
Citation: Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195
##### References:
 [1] J. Alves, "Statistical Analysis Ofnon-uniformly Expanding Dynamical Systems,", IMPA Mathematical Publications, 24 (2003). [2] M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions,, Proc. Amer. Math. Soc., 136 (2008), 677. doi: 10.1090/S0002-9939-07-09115-0. [3] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity,", Encyclopaedia of Mathematical Sciences, 102 (2004). [4] R. Bowen, Entropy-expansive maps,, Trans. A. M. S., 164 (1972), 323. doi: 10.1090/S0002-9947-1972-0285689-X. [5] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions,, Inventiones Math., 156 (2004), 119. doi: 10.1007/s00222-003-0335-2. [6] M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy, and subshift covers,, Forum Math., 14 (2002), 713. doi: 10.1515/form.2002.031. [7] D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions,, preprint, (). [8] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math., 171 (2010), 451. doi: 10.4007/annals.2010.171.451. [9] J. Buzzi, Intrinsic ergodicity for smooth interval maps,, Israel J. Math., 100 (1997), 125. doi: 10.1007/BF02773637. [10] J. Buzzi, T. Fisher, M. Sambarino and C. V\'asquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergod. Th. Dynamic. Systems, (). [11] W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits,, Ergod. Th. Dynamic. Systems, 25 (2005), 1115. doi: 10.1017/S0143385704000604. [12] L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms,, Discrete and Cont. Dynamic. Systems, 29 (2011), 1419. [13] T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions,, Inventiones Math., 176 (2009), 617. doi: 10.1007/s00222-008-0172-4. [14] T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems,, Inventiones Math., 160 (2005), 453. doi: 10.1007/s00222-004-0413-0. [15] N. Gourmelon, Adapted metrics for dominated splittings,, Ergod. Th. Dynamic. Systems, 27 (2007), 1839. [16] M. W. Hirsch, C.C. Pugh and M.Shub, "Invariant Manifolds,", Lecture Notes In Mathematics, 583 (1977). [17] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995). [18] G. Keller, "Equilibrium States in Ergodic Theory,", London Mathematical Society Student Texts, (1998). [19] G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies,, preprint, (). [20] M. Misiurewicz, Topological conditional entropy,, Studia Math., 55 (1976), 175. [21] M. J. Pacifico and J. L. Vieitez, Entropyexpansiveness and domination for surface diffeomorphisms,, Rev. Mat. Complut., 21 (2008), 293. [22] M. J. Pacifico and J. L. Vieitez, Robust entropy-expansiveness implies generic domination,, Nonlinearity, 23 (2010), 1971. doi: 10.1088/0951-7715/23/8/009. [23] V. A. Pliss, Analysis of the necessity of the conditions of Smale and Robbinfor structural stability of periodic systems of differentialequations,, Diff. Uravnenija, 8 (1972), 972. [24] R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center,, Topology Appl., 157 (2010), 29. [25] M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. [26] L. Wen, Homoclinic tangencies and dominated splittings,, Nonlinearity, 15 (2002), 1445. doi: 10.1088/0951-7715/15/5/306.

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##### References:
 [1] J. Alves, "Statistical Analysis Ofnon-uniformly Expanding Dynamical Systems,", IMPA Mathematical Publications, 24 (2003). [2] M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions,, Proc. Amer. Math. Soc., 136 (2008), 677. doi: 10.1090/S0002-9939-07-09115-0. [3] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity,", Encyclopaedia of Mathematical Sciences, 102 (2004). [4] R. Bowen, Entropy-expansive maps,, Trans. A. M. S., 164 (1972), 323. doi: 10.1090/S0002-9947-1972-0285689-X. [5] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions,, Inventiones Math., 156 (2004), 119. doi: 10.1007/s00222-003-0335-2. [6] M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy, and subshift covers,, Forum Math., 14 (2002), 713. doi: 10.1515/form.2002.031. [7] D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions,, preprint, (). [8] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math., 171 (2010), 451. doi: 10.4007/annals.2010.171.451. [9] J. Buzzi, Intrinsic ergodicity for smooth interval maps,, Israel J. Math., 100 (1997), 125. doi: 10.1007/BF02773637. [10] J. Buzzi, T. Fisher, M. Sambarino and C. V\'asquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergod. Th. Dynamic. Systems, (). [11] W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits,, Ergod. Th. Dynamic. Systems, 25 (2005), 1115. doi: 10.1017/S0143385704000604. [12] L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms,, Discrete and Cont. Dynamic. Systems, 29 (2011), 1419. [13] T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions,, Inventiones Math., 176 (2009), 617. doi: 10.1007/s00222-008-0172-4. [14] T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems,, Inventiones Math., 160 (2005), 453. doi: 10.1007/s00222-004-0413-0. [15] N. Gourmelon, Adapted metrics for dominated splittings,, Ergod. Th. Dynamic. Systems, 27 (2007), 1839. [16] M. W. Hirsch, C.C. Pugh and M.Shub, "Invariant Manifolds,", Lecture Notes In Mathematics, 583 (1977). [17] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995). [18] G. Keller, "Equilibrium States in Ergodic Theory,", London Mathematical Society Student Texts, (1998). [19] G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies,, preprint, (). [20] M. Misiurewicz, Topological conditional entropy,, Studia Math., 55 (1976), 175. [21] M. J. Pacifico and J. L. Vieitez, Entropyexpansiveness and domination for surface diffeomorphisms,, Rev. Mat. Complut., 21 (2008), 293. [22] M. J. Pacifico and J. L. Vieitez, Robust entropy-expansiveness implies generic domination,, Nonlinearity, 23 (2010), 1971. doi: 10.1088/0951-7715/23/8/009. [23] V. A. Pliss, Analysis of the necessity of the conditions of Smale and Robbinfor structural stability of periodic systems of differentialequations,, Diff. Uravnenija, 8 (1972), 972. [24] R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center,, Topology Appl., 157 (2010), 29. [25] M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. [26] L. Wen, Homoclinic tangencies and dominated splittings,, Nonlinearity, 15 (2002), 1445. doi: 10.1088/0951-7715/15/5/306.
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