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April  2012, 32(4): 1435-1447. doi: 10.3934/dcds.2012.32.1435

Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems

Pengfei Zhang 1,

1. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

Received  June 2010 Revised  August 2011 Published  October 2011

In [23] Xia introduced a simple dynamical density basis for partially hyperbolic sets of volume preserving diffeomorphisms. We apply the density basis to the study of the topological structure of partially hyperbolic sets. We show that if $\Lambda$ is a strongly partially hyperbolic set with positive volume, then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and the global unstable manifolds over ${\omega}(\Lambda^d)$.
    We give several applications of the dynamical density to partially hyperbolic maps that preserve some $acip$. We show that if $f$ is essentially accessible and $\mu$ is an $acip$ of $f$, then $\text{supp}(\mu)=M$, the map $f$ is transitive, and $\mu$-a.e. $x\in M$ has a dense orbit in $M$. Moreover if $f$ is accessible and center bunched, then either $f$ preserves a smooth measure or there is no $acip$ at all.
Keywords: Partially hyperbolic, $acip$ measure, positive volume, saturated., weak ergodicity, transitive, accessible, dynamical density basis.
Mathematics Subject Classification: Primary 37D30, 37D10; Secondary 37C40, 37D2.
Citation: Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435
References:
[1]

F. Abdenur and M. Viana, Flavors of partial hyperbolicity, preprint, 2008. Google Scholar

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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

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J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, in "Modern Dynamical Systems and Applications," Cambridge Univ. Press, Cambridge, (2004), 271-297.  Google Scholar

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

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R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

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M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Functional Anal. Appl., 9 (1975), 8-16. doi: 10.1007/BF01078168.  Google Scholar

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J. Pesin and M. Brin, Partially hyperbolic dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar

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M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

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K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math. (2), 171 (2010), 451-489.  Google Scholar

[11]

K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys., 108 (2002), 927-942. doi: 10.1023/A:1019779128351.  Google Scholar

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D. Dolgopyat and A Wilkinson, Stable accessibility is $C^1$ dense, in "Geometric Methods in Dynamics (II)", Astérisque 287 (2003), 33-60.  Google Scholar

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T. Fisher, "On the Structure of Hyperbolic Sets," Ph.D thesis, Northwestern University, 2004. Google Scholar

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J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.  Google Scholar

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H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

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N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynam. Sys., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.  Google Scholar

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M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[18]

V. Niţică and A. Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology, 40 (2001), 259-278. doi: 10.1016/S0040-9383(99)00060-9.  Google Scholar

[19]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013.  Google Scholar

[20]

C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176. doi: 10.1007/BF01390119.  Google Scholar

[21]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., 2007.  Google Scholar

[22]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, \arXiv{0809.4862}., ().   Google Scholar

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Z. Xia, Hyperbolic invariant sets with positive measures, Discrete Contin. Dyn. Syst., 15 (2006), 811-818. doi: 10.3934/dcds.2006.15.811.  Google Scholar

show all references

References:
[1]

F. Abdenur and M. Viana, Flavors of partial hyperbolicity, preprint, 2008. Google Scholar

[2]

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

[3]

A. Avila and J. Bochi, A generic $C^1$ map has no absolutely continuous invariant probability measure, Nonlinearity, 19 (2006), 2717-2725. doi: 10.1088/0951-7715/19/11/011.  Google Scholar

[4]

J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, in "Modern Dynamical Systems and Applications," Cambridge Univ. Press, Cambridge, (2004), 271-297.  Google Scholar

[5]

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

[6]

R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[7]

M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Functional Anal. Appl., 9 (1975), 8-16. doi: 10.1007/BF01078168.  Google Scholar

[8]

J. Pesin and M. Brin, Partially hyperbolic dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar

[9]

M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[10]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math. (2), 171 (2010), 451-489.  Google Scholar

[11]

K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys., 108 (2002), 927-942. doi: 10.1023/A:1019779128351.  Google Scholar

[12]

D. Dolgopyat and A Wilkinson, Stable accessibility is $C^1$ dense, in "Geometric Methods in Dynamics (II)", Astérisque 287 (2003), 33-60.  Google Scholar

[13]

T. Fisher, "On the Structure of Hyperbolic Sets," Ph.D thesis, Northwestern University, 2004. Google Scholar

[14]

J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.  Google Scholar

[15]

H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[16]

N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynam. Sys., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.  Google Scholar

[17]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[18]

V. Niţică and A. Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology, 40 (2001), 259-278. doi: 10.1016/S0040-9383(99)00060-9.  Google Scholar

[19]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013.  Google Scholar

[20]

C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176. doi: 10.1007/BF01390119.  Google Scholar

[21]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., 2007.  Google Scholar

[22]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, \arXiv{0809.4862}., ().   Google Scholar

[23]

Z. Xia, Hyperbolic invariant sets with positive measures, Discrete Contin. Dyn. Syst., 15 (2006), 811-818. doi: 10.3934/dcds.2006.15.811.  Google Scholar

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