Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on Tori
Abstract
We prove that every smooth action of Zk, k>1, on the (k+1)-dimensional torus homotopic to an action by hyperbolic linear maps preserves an absolutely continuous measure. This is a first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data.
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