Journal of Modern Dynamics (JMD)

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

Pages: 287 - 300, Issue 2, April 2007      doi:10.3934/jmd.2007.1.287

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Anatole Katok - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)
Federico Rodriguez Hertz - IMERL-Facultad de IngenierĂ­a, Universidad de la RepĂșblica, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay (email)

Abstract: 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.

Keywords:  measure rigidity, nonuniform hyperbolicity, $\mathbb{Z}^k$ actions.
Mathematics Subject Classification:  Primary: 37C40, 37D25, 37C85.

Received: May 2006;      Revised: December 2006;      Available Online: January 2007.