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 and Continuous Dynamical Systems, 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.

[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.

[3]

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

[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.

[5]

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

[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. 

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

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

[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.

[10]

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

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

[13]

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

[14]

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

[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.

[16]

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

[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.

[18]

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

[19]

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

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.

[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.

[3]

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

[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.

[5]

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

[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. 

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

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

[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.

[10]

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

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

[13]

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

[14]

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

[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.

[16]

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

[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.

[18]

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

[19]

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

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