Entropy-expansiveness for partially hyperbolic diffeomorphisms

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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

Abstract
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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 and 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.
[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.
[3]	C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity," Encyclopaedia of Mathematical Sciences, Math. Phys., 102, Springer-Verlag, Berlin, 2004.
[4]	R. Bowen, Entropy-expansive maps, Trans. A. M. S., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X.
[5]	M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Inventiones Math., 156 (2004), 119-161. 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), 713-757. doi: 10.1515/form.2002.031.
[7]	D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions, preprint, arXiv:0912.2018.
[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.
[9]	J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. 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, to appear.
[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.
[12]	L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete and Cont. Dynamic. Systems, 29 (2011), 1419-1441.
[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.
[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.
[15]	N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynamic. Systems, 27 (2007), 1839-1849.
[16]	M. W. Hirsch, C.C. Pugh and M.Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[17]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.
[18]	G. Keller, "Equilibrium States in Ergodic Theory," London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1998.
[19]	G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, preprint, arXiv:1012.0514.
[20]	M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.
[21]	M. J. Pacifico and J. L. Vieitez, Entropyexpansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.
[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.
[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.
[24]	R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center, Topology Appl., 157 (2010), 29-34.
[25]	M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.
[26]	L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. 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, Colóquio Brasileiro de Matemática, (IMPA), 2003.
[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.
[3]	C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity," Encyclopaedia of Mathematical Sciences, Math. Phys., 102, Springer-Verlag, Berlin, 2004.
[4]	R. Bowen, Entropy-expansive maps, Trans. A. M. S., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X.
[5]	M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Inventiones Math., 156 (2004), 119-161. 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), 713-757. doi: 10.1515/form.2002.031.
[7]	D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions, preprint, arXiv:0912.2018.
[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.
[9]	J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. 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, to appear.
[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.
[12]	L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete and Cont. Dynamic. Systems, 29 (2011), 1419-1441.
[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.
[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.
[15]	N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynamic. Systems, 27 (2007), 1839-1849.
[16]	M. W. Hirsch, C.C. Pugh and M.Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[17]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.
[18]	G. Keller, "Equilibrium States in Ergodic Theory," London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1998.
[19]	G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, preprint, arXiv:1012.0514.
[20]	M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.
[21]	M. J. Pacifico and J. L. Vieitez, Entropyexpansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.
[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.
[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.
[24]	R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center, Topology Appl., 157 (2010), 29-34.
[25]	M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.
[26]	L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306.

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