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

TL;DR

This paper establishes a uniform bound on the dynamics of certain endomorphisms of complex projective space, linking it to the Makienko conjecture and properties of Julia sets in complex dynamics.

Contribution

It introduces a new a priori bound for endomorphisms of bCP^k under specific measure conditions, connecting it to the Makienko conjecture in dimension one.

Findings

Derived a locally uniform a priori bound for the dynamics of endomorphisms.

Connected measure conditions on Fatou components to the Makienko conjecture.

Provided Diophantine-type estimates for dynamics on singular domains.

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 $f$.