Quantitative logarithmic equidistribution of the crucial measures

Abstract

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let $$\phi \in K(z)$$ with $$\deg (\phi )\ge 2$$ . Recently, Rumely introduced a family of discrete probability measures $$\{\nu _{\phi ^n}\}$$ on the Berkovich line $$\mathbf{P }^1_{\text {K}}$$ over K which carry information about the reduction of conjugates of $$\phi $$ . In a previous article, the author showed that the measures $$\nu _{\phi ^n}$$ converge weakly to the canonical measure $$\mu _\phi $$ . In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of $$\mathbf{P }^1_{\text {K}}$$ . These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to $$\nu _{\phi ^n}$$ converge to the potential function attached to $$\mu _\phi $$ , as well as an approximation result for the Lyapunov exponent of $$\phi $$ .