  Let $f$ be a post-critically finite endomorphism (PCF map for short) on
$\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support
of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus
J_2$ is contained in the union of the (finitely many) basins of critical
component cycles and stable manifolds of sporadic super-saddle cycles. 2. For
every $x\in J_2$ which is not contained in the stable manifold of a sporadic
super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means
that the super-saddle cycle is not contained in a critical component cycle.
Under the additional assumption that all branches of $PC(f)$ are smooth and
intersect transversally, we show that there is no sporadic super-saddle cycle.
Thus in this case $J_1\setminus J_2$ is contained in the union of the basins of
critical component cycles, and for every $x\in J_2$ there is no Fatou disk
containing $x$. As consequences of our result: 1.We answer some questions of
Fornaess-Sibony about the non-wandering set for PCF maps on $\mathbb{P}^2$ with
no sporadic super-saddle cycles. 2. We give a new proof of de Th\'elin's
laminarity of the Green current in $J_1\setminus J_2$ for PCF maps on
$\mathbb{P}^2$. 3. We show that for PCF maps on $\mathbb{P}^2$ an invariant
compact set is expanding if and only if it does not contain critical points,
and we obtain characterizations of PCF maps on $\mathbb{P}^2$ which are
expanding on $J_2$ or satisfy Axiom A.
