# American Institute of Mathematical Sciences

December  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, 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, Colóquio Brasileiro de Matemática, (IMPA), 2003.  Google Scholar [2] M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc., 136 (2008), 677-686. doi: 10.1090/S0002-9939-07-09115-0.  Google Scholar [3] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity," Encyclopaedia of Mathematical Sciences, Math. Phys., 102, Springer-Verlag, Berlin, 2004.  Google Scholar [4] R. Bowen, Entropy-expansive maps, Trans. A. M. S., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar [5] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Inventiones Math., 156 (2004), 119-161. doi: 10.1007/s00222-003-0335-2.  Google Scholar [6] M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy, and subshift covers, Forum Math., 14 (2002), 713-757. doi: 10.1515/form.2002.031.  Google Scholar [7] D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions,, preprint, ().   Google Scholar [8] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.  Google Scholar [9] J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. doi: 10.1007/BF02773637.  Google Scholar [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, ().   Google Scholar [11] W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138. doi: 10.1017/S0143385704000604.  Google Scholar [12] L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete and Cont. Dynamic. Systems, 29 (2011), 1419-1441.  Google Scholar [13] T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions, Inventiones Math., 176 (2009), 617-636. doi: 10.1007/s00222-008-0172-4.  Google Scholar [14] T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems, Inventiones Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.  Google Scholar [15] N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynamic. Systems, 27 (2007), 1839-1849.  Google Scholar [16] M. W. Hirsch, C.C. Pugh and M.Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar [18] G. Keller, "Equilibrium States in Ergodic Theory," London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1998.  Google Scholar [19] G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies,, preprint, ().   Google Scholar [20] M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.  Google Scholar [21] M. J. Pacifico and J. L. Vieitez, Entropyexpansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.  Google Scholar [22] M. J. Pacifico and J. L. Vieitez, Robust entropy-expansiveness implies generic domination, Nonlinearity, 23 (2010), 1971-1990. doi: 10.1088/0951-7715/23/8/009.  Google Scholar [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), 972-983.  Google Scholar [24] R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center, Topology Appl., 157 (2010), 29-34.  Google Scholar [25] M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar [26] L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306.  Google Scholar

show all references

##### References:
 [1] J. Alves, "Statistical Analysis Ofnon-uniformly Expanding Dynamical Systems," IMPA Mathematical Publications, 24, Colóquio Brasileiro de Matemática, (IMPA), 2003.  Google Scholar [2] M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc., 136 (2008), 677-686. doi: 10.1090/S0002-9939-07-09115-0.  Google Scholar [3] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity," Encyclopaedia of Mathematical Sciences, Math. Phys., 102, Springer-Verlag, Berlin, 2004.  Google Scholar [4] R. Bowen, Entropy-expansive maps, Trans. A. M. S., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar [5] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Inventiones Math., 156 (2004), 119-161. doi: 10.1007/s00222-003-0335-2.  Google Scholar [6] M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy, and subshift covers, Forum Math., 14 (2002), 713-757. doi: 10.1515/form.2002.031.  Google Scholar [7] D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions,, preprint, ().   Google Scholar [8] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.  Google Scholar [9] J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. doi: 10.1007/BF02773637.  Google Scholar [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, ().   Google Scholar [11] W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138. doi: 10.1017/S0143385704000604.  Google Scholar [12] L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete and Cont. Dynamic. Systems, 29 (2011), 1419-1441.  Google Scholar [13] T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions, Inventiones Math., 176 (2009), 617-636. doi: 10.1007/s00222-008-0172-4.  Google Scholar [14] T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems, Inventiones Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.  Google Scholar [15] N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynamic. Systems, 27 (2007), 1839-1849.  Google Scholar [16] M. W. Hirsch, C.C. Pugh and M.Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar [18] G. Keller, "Equilibrium States in Ergodic Theory," London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1998.  Google Scholar [19] G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies,, preprint, ().   Google Scholar [20] M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.  Google Scholar [21] M. J. Pacifico and J. L. Vieitez, Entropyexpansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.  Google Scholar [22] M. J. Pacifico and J. L. Vieitez, Robust entropy-expansiveness implies generic domination, Nonlinearity, 23 (2010), 1971-1990. doi: 10.1088/0951-7715/23/8/009.  Google Scholar [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), 972-983.  Google Scholar [24] R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center, Topology Appl., 157 (2010), 29-34.  Google Scholar [25] M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar [26] L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306.  Google Scholar
 [1] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006 [2] Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421 [3] Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419 [4] Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087 [5] David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873 [6] Mike Boyle, Tomasz Downarowicz. Symbolic extension entropy: $c^r$ examples, products and flows. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 329-341. doi: 10.3934/dcds.2006.16.329 [7] Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27 [8] Vaughn Climenhaga, Yakov Pesin, Agnieszka Zelerowicz. Equilibrium measures for some partially hyperbolic systems. Journal of Modern Dynamics, 2020, 16: 155-205. doi: 10.3934/jmd.2020006 [9] Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469 [10] Jacek Serafin. A faithful symbolic extension. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1051-1062. doi: 10.3934/cpaa.2012.11.1051 [11] Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341 [12] Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789 [13] David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253 [14] Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555 [15] Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004 [16] Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219 [17] Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237 [18] Martín Sambarino, José L. Vieitez. Robustly expansive homoclinic classes are generically hyperbolic. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1325-1333. doi: 10.3934/dcds.2009.24.1325 [19] Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487 [20] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435

2020 Impact Factor: 1.392