Geometric and measure-theoretical structures of maps with mostly   contracting center
  We show that every diffeomorphism with mostly contracting center direction exhibits a geometric-combinatorial structure, which we call \emph{skeleton}, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how the physical measure bifurcate as the diffeomorphism changes. In particular, we use this to construct examples with any given number of physical measures, with basins densely intermingled, and to analyse how these measures collapse into each other - through explosions of their basins - as the dynamics varies. This theory also allows us to prove that, in the absence of collapses, the basins are continuous functions of the diffeomorphism. 
Dmitry Dolgopyat, Marcelo Viana, Jiagang Yang
math.DS math-ph math.CO math.MP