Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori
Boris Kalinin - Department of Mathematics, University of South Alabama, Mobile, AL 36688, United States (email) Abstract:
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.
Keywords: measure rigidity, nonuniform hyperbolicity, $\mathbb{Z}^k$ actions.
Received: March 2006; Available Online: October 2006. |