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

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

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 …