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

Boris Kalinin, Anatole Katok

Published 2006-02-08Version 1

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.