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

Abstract

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 $f$.

arXiv comment6 pages. (v4) The complex part in arXiv:1805.07668v1 is marged with the title modification. We point out that in one dimension, the assumption in the main theorem is related to the impossibility for the Julia set set of $f$ to be the boundary of lakes of Wada and the Makienko conjecture, but those two problems are still open (v3) one reference is added (v2) more comments are added in Remark 2.2