American Institute of Mathematical Sciences

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

Entropy-expansiveness for partially hyperbolic diffeomorphisms

Lorenzo J. Díaz 1, , Todd Fisher 2, , M. J. Pacifico 3, and  José L. Vieitez 4,

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.
Keywords: symbolic extension., Entropy-expansive, dominated splitting, equilibrium state, partially hyperbolic.
Mathematics Subject Classification: 37D30, 37C05, 37B1.
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

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy