  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.
