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Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on Tori

Boris Kalinin·Anatole Katok
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Abstract

We prove that every smooth action of Z^k, 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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Boris Kalinin, Anatole Katok. 2006-02-08. Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on Tori. https://arxiv.org/abs/math/0602176

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