Nonarchimedean Green functions and dynamics on projective space

TL;DR

This paper studies nonarchimedean Green functions for morphisms on projective space, proving their continuity, the characterization of Fatou sets, and properties of iterates in a nonarchimedean setting.

Contribution

It establishes the Holder continuity of Green functions, characterizes Fatou sets via local constancy of Green functions, and analyzes the Lipschitz behavior of iterates over nonarchimedean fields.

Findings

Green function G_F is Holder continuous on P^N(K)

Fatou set equals points where G_F is locally constant

Iterates of F are locally Lipschitz on the Fatou set

Abstract

Let F: P^N_K --> P^N_K be a morphism of degree d > 1 defined over a field K that is algebraically closed and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function G_F associated to F is Holder continuous on P^N(K) and that the Fatou set F is equal to the set of points at which G_F is locally constant. Further, G_F vanishes precisely on the set of points P such that F has good reduction at every point in the forward orbit of P. We also prove that the iterates of F are locally uniformly Lipschitz on the Fatou set of F.