Iteration and the Minimal Resultant

Abstract

Let $K$ be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let $\phi \in K(z)$ have degree $d\ge 2$. We characterize maps for which the minimal resultant of an iterate ${\phi }^n$ is given by a simple formula in terms of $d$, $n$, and the minimal resultant of $\phi$. We show that such maps are precisely those with reduction outside of an indeterminacy locus $I(d)$ and which also have semi-stable reduction for every iterate ${\phi }^n$. We give two equivalent ways of describing such maps, one measure theoretic and the other in terms of the moduli space ${\mathcal{M}}_d$ of degree $d$ rational maps. As an application, we are able to give an explicit formula for the minimal value of the diagonal Arakelov-Green's function of a map satisfying the conditions of the main theorem. We illustrate our results with some explicit calculations in the case of the Lattès maps.

39 pages. Comments welcome!

References in corpus (1)

• The Minimal Resultant Locus