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

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.
Citation: Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 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

[2]

J. Alves and V. Pinheiro, Topological structure of (partially) hyperbolic sets with positive volume,, Trans. Amer. Math. Soc., 360 (2008), 5551.  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.  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, (2004), 271.   Google Scholar

[5]

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

[6]

R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms,", Lecture Notes in Mathematics, 470 (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.  doi: 10.1007/BF01078168.  Google Scholar

[8]

J. Pesin and M. Brin, Partially hyperbolic dynamical systems,, (Russian), 38 (1974), 170.   Google Scholar

[9]

M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (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.   Google Scholar

[11]

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

[12]

D. Dolgopyat and A Wilkinson, Stable accessibility is $C^1$ dense,, in, 287 (2003), 33.   Google Scholar

[13]

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

[14]

J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301.  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, (1981).   Google Scholar

[16]

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

[17]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (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.  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.  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.  doi: 10.1007/BF01390119.  Google Scholar

[21]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics,, in, 51 (2007), 35.   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.  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.  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.  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, (2004), 271.   Google Scholar

[5]

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

[6]

R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms,", Lecture Notes in Mathematics, 470 (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.  doi: 10.1007/BF01078168.  Google Scholar

[8]

J. Pesin and M. Brin, Partially hyperbolic dynamical systems,, (Russian), 38 (1974), 170.   Google Scholar

[9]

M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (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.   Google Scholar

[11]

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

[12]

D. Dolgopyat and A Wilkinson, Stable accessibility is $C^1$ dense,, in, 287 (2003), 33.   Google Scholar

[13]

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

[14]

J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301.  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, (1981).   Google Scholar

[16]

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

[17]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (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.  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.  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.  doi: 10.1007/BF01390119.  Google Scholar

[21]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics,, in, 51 (2007), 35.   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.  doi: 10.3934/dcds.2006.15.811.  Google Scholar

[1]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233

[2]

Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63

[3]

Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789

[4]

Keith Burns, Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Anna Talitskaya, Raúl Ures. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 75-88. doi: 10.3934/dcds.2008.22.75

[5]

R.E. Showalter, Ning Su. Partially saturated flow in a poroelastic medium. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 403-420. doi: 10.3934/dcdsb.2001.1.403

[6]

Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1

[7]

Yunhua Zhou. The local $C^1$-density of stable ergodicity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2621-2629. doi: 10.3934/dcds.2013.33.2621

[8]

François Ledrappier, Seonhee Lim. Volume entropy of hyperbolic buildings. Journal of Modern Dynamics, 2010, 4 (1) : 139-165. doi: 10.3934/jmd.2010.4.139

[9]

Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287

[10]

Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228

[11]

David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253

[12]

Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004

[13]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259

[14]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74

[15]

Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037

[16]

Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419

[17]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271

[18]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

[19]

Dmitry Dolgopyat. The work of Federico Rodriguez Hertz on ergodicity of dynamical systems. Journal of Modern Dynamics, 2016, 10: 175-189. doi: 10.3934/jmd.2016.10.175

[20]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020041

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]