Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data
Anatole Katok - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (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.
Received: May 2006; Revised: December 2006; Available Online: January 2007. |