  It is a classical result in complex dynamics of one variable that the Fatou
set for a critically finite map on $\mathbf{P}^1$ consists of only basins of
attraction for superattracting periodic points. In this paper we deal with
critically finite maps on $\mathbf{P}^k$. We show that the Fatou set for a
critically finite map on $\mathbf{P}^2$ consists of only basins of attraction
for superattracting periodic points. We also show that the Fatou set for a
$k-$critically finite map on $\mathbf{P}^k$ is empty.
