American Institute of Mathematical Sciences

Advanced
April  2008, 2(2): 187-208. doi: 10.3934/jmd.2008.2.187

Partial hyperbolicity and ergodicity in dimension three

Federico Rodriguez Hertz 1, , María Alejandra Rodriguez Hertz 2, and  Raúl Ures 2,

1. 

IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay
2. 

IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay, Uruguay

Received  March 2007 Revised  November 2007 Published  January 2008

In [15] the authors proved the Pugh–Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e., stably ergodic diffeomorphisms are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the first examples of manifolds in which all conservative partially hyperbolic diffeomorphisms are ergodic.
Keywords: partial hyperbolicity, laminations., accessibility property, ergodicity.
Mathematics Subject Classification: Primary: 37D30, Secondary: 37A2.
Citation: Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187
[1]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012
[2]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[3]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527
[4]

Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227
[5]

Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641
[6]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[7]

Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81
[8]

Jana Rodriguez Hertz, Carlos H. Vásquez. Structure of accessibility classes. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4653-4664. doi: 10.3934/dcds.2020196
[9]

Alexander Blokh, Lex Oversteegen, Ross Ptacek, Vladlen Timorin. Laminations from the main cubioid. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4665-4702. doi: 10.3934/dcds.2016003
[10]

Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029
[11]

Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1175-1189. doi: 10.3934/dcds.2011.29.1175
[12]

Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819
[13]

Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845
[14]

Meiyu Su. True laminations for complex Hènon maps. Conference Publications, 2003, 2003 (Special) : 834-841. doi: 10.3934/proc.2003.2003.834
[15]

Alexander Blokh, Clinton Curry, Lex Oversteegen. Density of orbits in laminations and the space of critical portraits. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2027-2039. doi: 10.3934/dcds.2012.32.2027
[16]

Alexander Blokh. Necessary conditions for the existence of wandering triangles for cubic laminations. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 13-34. doi: 10.3934/dcds.2005.13.13
[17]

Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
[18]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
[19]

Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control & Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002
[20]

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, 2008, 22 (1&2) : 75-88. doi: 10.3934/dcds.2008.22.75

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences