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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

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