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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 weconsider the family of functions $\textrm{ordRes}_{\phi^n}(x)$, which measurethe resultant of $\phi^n$ at points $x$ in $\textbf{P}^1_{\textrm{K}}$, theBerkovich projective line, and show that they converge locally uniformly to thediagonal values of the Arakelov-Green's function $g_{\mu_{\phi}}(x,x)$ attachedto the canonical measure of $\phi$. Following this, we are able to prove anequidistribution result for Rumely's crucial measures $\nu_{\phi^n}$, each ofwhich is a probability measure supported at finitely many points whose weightsare determined by dynamical properties of $\phi$.

An Equidistribution Result For Dynamical Systems on the Berkovich Projective Line