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
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