Resultant measures and minimal resultant loci for non-archimedean polynomial dynamics

Abstract

We compute the resultant measures for iterations $P^j$, $j\ge 1$, of a polynomial $P$ of degree $>1$ on the $n$-th level Trucco's trees ${\Gamma }_n$, $n\ge 0$, in the Berkovich projective line over a non-archimedean field and also determine their barycenters. As applications, we study the asymptotic of those barycenters as $n\to \infty$, and establish a uniform stationarity of Rumely's minimal resultant loci of $P^j$ or equivalently that of the potential semistable reduction loci of $P^j$ as $j\to \infty$. We also establish several equidistribution results for the resultant measures themselves as $n\to \infty$.

37 pages, 1 figure. The terminologies are made more informative and the title is slightly modified. The presentation is improved, and the Hilbert-Mumford GIT is dispensed with in all the proofs but for in reformulating the identity (1.1) as Theorem 4