A general endomorphism of ${\mathbb{P}}^k$ has trivial iterated centralizer

Abstract

Fix integers $k,d\ge 2$. Let ${\mathrm{End}}_d^k$ denote the parameter space of holomorphic endomorphisms of ${\mathbb{P}}^k$ of algebraic degree $d$. We prove that there exists a dense Zariski open subset $U_{d,k}\subset {\mathrm{End}}_d^k$, defined over $\mathbb{Q}$, such that, for every $f\in U_{d,k}(\mathbb{C})$, every nonconstant endomorphism $g:{\mathbb{P}}^k\to {\mathbb{P}}^k$ commuting with an iterate of $f$ is itself an iterate of $f$. An analogous dense Zariski open subset exists for regular polynomial endomorphisms of ${\mathbb{C}}^k$. The proof uses finite-level monodromy and its action on the rooted preimage tree of a point outside the branch locus of an iterate of $f$. As an application, we show that, for a general $f\in {\mathrm{End}}_d^k$, an irreducible hypersurface of ${\mathbb{P}}^k$ is $f$-special in the sense of Ghioca--Tucker and DeMarco--Mavraki if and only if it is $f$-preperiodic.

34 pages, comments are welcome