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Journal of Modern Dynamics (JMD)
 

Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori

Pages: 123 - 146, Issue 1, January 2007      doi:10.3934/jmd.2007.1.123

 
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Boris Kalinin - Department of Mathematics, University of South Alabama, Mobile, AL 36688, United States (email)
Anatole Katok - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, 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.
    We also show that both ergodic and geometric properties of such a measure are very close to the corresponding properties of the Lebesgue measure with respect to the linear action $\a_0$.

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

Received: March 2006;      Available Online: October 2006.