  We prove a global fixed point theorem for the centralizer of a homeomorphism
of the two dimensional disk $D$ that has attractor-repeller dynamics on the
boundary with at least two attractors and two repellers. As one application, we
show that there is a finite index subgroup of the centralizer of a
pseudo-Anosov homeomorphism with infinitely many global fixed points. As
another application we give an elementary proof of Morita's Theorem, that the
mapping class group of a closed surface $S$ of genus $g$ does not lift to the
group of diffeormorphisms of $S$ and we improve the lower bound for $g$ from 5
to 3.
