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 $\varphi \in K(z)$ with $\text{deg}(\varphi )\ge 2$. In this paper we consider the family of functions ${\text{ordRes}}_{{\varphi }^n}(x)$, which measure the resultant of ${\varphi }^n$ at points $x$ in ${\text{P}}_{\text{K}}^1$, the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov-Green's function $g_{{\mu }_{\varphi }}(x,x)$ attached to the canonical measure of $\varphi$. Following this, we are able to prove an equidistribution result for Rumely's crucial measures ${\nu }_{{\varphi }^n}$, each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of $\varphi$.

To appear in Journal of Number Theory

References in corpus (1)

• The Geometry of the Minimal Resultant Locus

Cited by in corpus (4)

• The Geometry of the Minimal Resultant Locus
• Iteration and the Minimal Resultant
• Resultant measures and minimal resultant loci for non-archimedean polynomial dynamics
• Geometric formulas on Rumely's weight function and crucial measure in non-archimedean dynamics