# American Institute of Mathematical Sciences

April  2007, 1(2): 287-300. doi: 10.3934/jmd.2007.1.287

## Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data

 1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 2 IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay

Received  May 2006 Revised  December 2006 Published  January 2007

Every $C^2$ action $\a$ of $\mathbb{Z}k$, $k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to the corresponding elements of an action $\ao$ by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between $\a$ and $\a_0$. This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for $\a$ for which the eigenvalues for $\a$ and $\a_0$ coincide. We describe some nontrivial examples of actions of this kind.
Citation: 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
 [1] 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 [2] 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 [3] 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 [4] Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765 [6] Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2010, 27 (2) : 609-615. doi: 10.3934/dcds.2010.27.609 [8] 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 [9] 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 [10] 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 [11] 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 [12] Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130 [13] 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 [14] Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55 [15] Changguang Dong. Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4231-4238. doi: 10.3934/dcds.2017180 [16] 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 [17] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68 [18] 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 [19] Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269 [20] A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

2021 Impact Factor: 0.641