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.

show all references

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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