Measure rigidity beyond uniform hyperbolicity: Invariant Measures for   Cartan actions on Tori
  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. 
Boris Kalinin, Anatole Katok