An Equidistribution Result For Dynamical Systems on the Berkovich Projective Line

TL;DR

This paper proves that certain dynamical functions on the Berkovich projective line converge to the Arakelov-Green's function, leading to an equidistribution result for measures associated with iterates of a rational function over a non-Archimedean field.

Contribution

It establishes the convergence of ordRes functions to the Arakelov-Green's function and proves an equidistribution theorem for Rumely's crucial measures in non-Archimedean dynamics.

Findings

Convergence of ordRes functions to the Arakelov-Green's function.

Proof of equidistribution for Rumely's crucial measures.

Application to non-Archimedean dynamical systems.

Abstract

Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\phi\in K(z)$ with $\textrm{deg}(\phi) \geq 2$. In this paper we consider the family of functions $\textrm{ordRes}_{\phi^n}(x)$, which measure the resultant of $\phi^n$ at points $x$ in $\textbf{P}^1_{\textrm{K}}$, the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov-Green's function $g_{\mu_{\phi}}(x,x)$ attached to the canonical measure of $\phi$. Following this, we are able to prove an equidistribution result for Rumely's crucial measures $\nu_{\phi^n}$, each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of $\phi$.