June  2018, 38(6): 2717-2729. doi: 10.3934/dcds.2018114

Partially hyperbolic sets with a dynamically minimal lamination

Departamento de Matemática - Instituto Tecnológico de Aeronáutica (ITA) - Praça Marechal Eduardo Gomes, 50 - Vila das Acacias, São José dos Campos, CEP 12228-900, SP, Brazil

Received  March 2017 Revised  December 2017 Published  April 2018

We study partially hyperbolic sets of $C^1$-diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely.

We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to $C^1$-generic/robustly transitive attractors with one-dimensional center bundle.

Citation: Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114
References:
[1]

F. Abdenur, Attractors of generic diffeomorphisms are persistent, Nonlinearity, 16 (2003), 301-311. doi: 10.1088/0951-7715/16/1/318. Google Scholar

[2]

F. AbdenurC. Bonatti and L. Díaz, Non-wandering sets with non-empty interior, Nonlinearity, 17 (2004), 175-191. doi: 10.1088/0951-7715/17/1/011. Google Scholar

[3]

F. Abdenur and S. Crovisier, Transitivity and topological mixing for C1 diffeomorphisms, Essays in Mathematics and Its Applications, Springer, Heidelberg, (2012), 1-16. Google Scholar

[4]

J. Alves and V. Pinheiro, Topological structure of partially hyperbolic sets with positive volume, Trans. Amer. Math. Soc., 360 (2008), 5551-5569. doi: 10.1090/S0002-9947-08-04484-X. Google Scholar

[5]

C. Bonatti and S. Crovisier, Recurrence et genéricité, Invent. Math., 1 (2002), 513-541. Google Scholar

[6]

C. BonattiL. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541. Google Scholar

[7] C. BonattiL. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer-Verlag, Berlin, 2005. Google Scholar
[8]

C. BonattiS. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergod. Th. & Dynam. Sys., 27 (2007), 1473-1508. Google Scholar

[9]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193. doi: 10.1007/BF02810585. Google Scholar

[10]

R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204. doi: 10.1007/BF01389849. Google Scholar

[11] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Berlin-New York, 1975. Google Scholar
[12]

M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Prilozen, 9 (1975), 9-19. Google Scholar

[13]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, Amer. Math. Soc., Providence, R. I., 1978. Google Scholar

[14]

T. Fisher, Hyperbolic sets with non-empty interior, Discrete Contin. Dyn. Syst., 15 (2006), 433-446. doi: 10.3934/dcds.2006.15.433. Google Scholar

[15]

F. Hertz, M. Hertz and R. Ures, Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, 103-109, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007. Google Scholar

[16]

R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021. Google Scholar

[17]

F. Nobili, Minimality of one invariant foliation for partially hyperbolic attractors, Nonlinearity, 28 (2015), 1897-1918. doi: 10.1088/0951-7715/28/6/1897. Google Scholar

[18]

R. Potrie, Generic bi-Lyapunov stable homoclinic classes, Nonlinearity, 23 (2010), 1631-1649. doi: 10.1088/0951-7715/23/7/006. Google Scholar

[19]

S. Smale, Diffeomorphisms with many periodic points, Bull. Am. Math. Soc., 73 (1967), 747-817. Google Scholar

show all references

References:
[1]

F. Abdenur, Attractors of generic diffeomorphisms are persistent, Nonlinearity, 16 (2003), 301-311. doi: 10.1088/0951-7715/16/1/318. Google Scholar

[2]

F. AbdenurC. Bonatti and L. Díaz, Non-wandering sets with non-empty interior, Nonlinearity, 17 (2004), 175-191. doi: 10.1088/0951-7715/17/1/011. Google Scholar

[3]

F. Abdenur and S. Crovisier, Transitivity and topological mixing for C1 diffeomorphisms, Essays in Mathematics and Its Applications, Springer, Heidelberg, (2012), 1-16. Google Scholar

[4]

J. Alves and V. Pinheiro, Topological structure of partially hyperbolic sets with positive volume, Trans. Amer. Math. Soc., 360 (2008), 5551-5569. doi: 10.1090/S0002-9947-08-04484-X. Google Scholar

[5]

C. Bonatti and S. Crovisier, Recurrence et genéricité, Invent. Math., 1 (2002), 513-541. Google Scholar

[6]

C. BonattiL. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541. Google Scholar

[7] C. BonattiL. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer-Verlag, Berlin, 2005. Google Scholar
[8]

C. BonattiS. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergod. Th. & Dynam. Sys., 27 (2007), 1473-1508. Google Scholar

[9]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193. doi: 10.1007/BF02810585. Google Scholar

[10]

R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204. doi: 10.1007/BF01389849. Google Scholar

[11] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Berlin-New York, 1975. Google Scholar
[12]

M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Prilozen, 9 (1975), 9-19. Google Scholar

[13]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, Amer. Math. Soc., Providence, R. I., 1978. Google Scholar

[14]

T. Fisher, Hyperbolic sets with non-empty interior, Discrete Contin. Dyn. Syst., 15 (2006), 433-446. doi: 10.3934/dcds.2006.15.433. Google Scholar

[15]

F. Hertz, M. Hertz and R. Ures, Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, 103-109, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007. Google Scholar

[16]

R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021. Google Scholar

[17]

F. Nobili, Minimality of one invariant foliation for partially hyperbolic attractors, Nonlinearity, 28 (2015), 1897-1918. doi: 10.1088/0951-7715/28/6/1897. Google Scholar

[18]

R. Potrie, Generic bi-Lyapunov stable homoclinic classes, Nonlinearity, 23 (2010), 1631-1649. doi: 10.1088/0951-7715/23/7/006. Google Scholar

[19]

S. Smale, Diffeomorphisms with many periodic points, Bull. Am. Math. Soc., 73 (1967), 747-817. Google Scholar

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