  We show that in the family of degree $d\geq 2$ rational maps of the Riemann
sphere, the closure of strictly postcritically finite maps contains a
(relatively) Baire generic subset of maps displaying maximal non-statistical
behavior: for a map $f$ in this generic subset, the set of accumulation points
of the sequence of empirical measures of almost every point in the phase space
is the largest possible one that is, the set of all $f$-invariant measures. The
proofs is based on a transversality argument which allows us to control the
behavior of the orbits of critical points for maps close to strictly
postcritically finite rational maps and also a new concept developed in the
author's PhD thesis, that we call statistical bifurcation.
