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Let $f$ be an endomorphism of $\mathbb{C}\mathbb{P}^k$ of degree $>1$, andassume 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 theboundary of $W$ if and only if $f^{-p}(W)=W$. Then we obtain a locally uniforma priori bound of the dynamics of $f$, which in particular yields aDiophantine-type estimate of the dynamics of $f$ on its domaines singuliers. Wealso point out that in the case of $k=1$, the statement of our assumption isrelated to both the impossibility for the Julia set of $f$ to be the boundaryof lakes of Wada and the so called Makienko conjecture on the non-emptiness ofthe residual Julia set of $f$.

An a priori bound of endomorphisms of $\mathbb{C}\mathbb{P}^k$ and a remark on the Makienko conjecture in dimension one