  Let $f$ be an endomorphism of $\mathbb{C}\mathbb{P}^k$ of degree $>1$, and
assume that for any cyclic Fatou component $W$ of $f$ having a period
$p\in\mathbb{N}$, the equilibrium measure $\mu_f$ has a positive charge on the
boundary of $W$ if and only if $f^{-p}(W)=W$. Then we obtain a locally uniform
a priori bound of the dynamics of $f$, which in particular yields a
Diophantine-type estimate of the dynamics of $f$ on its domaines singuliers. We
also point out that in the case of $k=1$, the statement of our assumption is
related to both the impossibility for the Julia set of $f$ to be the boundary
of lakes of Wada and the so called Makienko conjecture on the non-emptiness of
the residual Julia set of $f$.
