American Institute of Mathematical Sciences

Advanced
January  2007, 1(1): 123-146. doi: 10.3934/jmd.2007.1.123

Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori

Boris Kalinin 1, and  Anatole Katok 2,

1. 

Department of Mathematics, University of South Alabama, Mobile, AL 36688, United States
2. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received  March 2006 Published  October 2006

We prove that every smooth action $\a$ of $\mathbb{Z}^k,k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to corresponding elements of an action $\a_0$ by hyperbolic linear maps preserves an absolutely continuous measure. This is the first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data.
    We also show that both ergodic and geometric properties of such a measure are very close to the corresponding properties of the Lebesgue measure with respect to the linear action $\a_0$.
Keywords: $\mathbb{Z}^k$ actions., measure rigidity, nonuniform hyperbolicity.
Mathematics Subject Classification: Primary: 37C40,37D25, 37C8.
Citation: Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123
[1]

Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207
[2]

Boris Kalinin, Anatole Katok and Federico Rodriguez Hertz. New progress in nonuniform measure and cocycle rigidity. Electronic Research Announcements, 2008, 15: 79-92. doi: 10.3934/era.2008.15.79
[3]

Anatole Katok, Federico Rodriguez Hertz. Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data. Journal of Modern Dynamics, 2007, 1 (2) : 287-300. doi: 10.3934/jmd.2007.1.287
[4]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
[5]

Jonathan Meddaugh, Brian E. Raines. The structure of limit sets for $\mathbb{Z}^d$ actions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765
[6]

Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509
[7]

Anatole Katok, Federico Rodriguez Hertz. Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 609-615. doi: 10.3934/dcds.2010.27.609
[8]

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
[9]

Jana Rodriguez Hertz. Genericity of nonuniform hyperbolicity in dimension 3. Journal of Modern Dynamics, 2012, 6 (1) : 121-138. doi: 10.3934/jmd.2012.6.121
[10]

Huimin Zheng, Xuejun Guo, Hourong Qin. The Mahler measure of $ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $. Electronic Research Archive, 2020, 28 (1) : 103-125. doi: 10.3934/era.2020007
[11]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487
[12]

Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571
[13]

Changguang Dong. Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4231-4238. doi: 10.3934/dcds.2017180
[14]

Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55
[15]

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
[16]

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

Giovanni Forni. A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle. Journal of Modern Dynamics, 2011, 5 (2) : 355-395. doi: 10.3934/jmd.2011.5.355
[18]

Helena Rifà-Pous, Josep Rifà, Lorena Ronquillo. $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography. Advances in Mathematics of Communications, 2011, 5 (3) : 425-433. doi: 10.3934/amc.2011.5.425
[19]

Makoto Araya, Masaaki Harada, Hiroki Ito, Ken Saito. On the classification of $\mathbb{Z}_4$-codes. Advances in Mathematics of Communications, 2017, 11 (4) : 747-756. doi: 10.3934/amc.2017054
[20]

Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences