An Equidistribution Result For Dynamical Systems on the Berkovich Projective Line

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 $\phi$.