Selberg and Brolin on value distribution of complex dynamics

TL;DR

This paper improves the quantification of key equidistribution theorems in complex dynamics, specifically for rational functions, by refining previous arguments and extending results under certain hypotheses.

Contribution

It demonstrates that earlier methods already provide better quantification of the equidistribution theorems and extends these results to Lyubich's theorem under an exponentwise hypothesis H.

Findings

Enhanced quantification of the Brolin-Lyubich-Freire--Lopes--Ma ñé theorem

Extension of Lyubich's theorem under hypothesis H

Simplification of previous argumentation for better results

Abstract

The Brolin-Lyubich-Freire--Lopes--Ma\~n\'e equidistribution theorem for iterated preimages of a given non-exceptional value and Lyubich's periodic point version of it are foundational in the study of dynamics of rational functions of degree more than one on the complex projective line, and Drasin and the author studied a quantification of the former in a formalism of Nevanlinna theory or more specifically with the aid of Selberg's theorem. In this paper, we point out that the argument in that previous study have already yielded a better quantification of the Brolin-Lyubich-Freire--Lopes--Ma\~n\'e equidistribution theorem, and also point out that a similar argument also yields a quantification of Lyubich's theorem under an exponentwise version of the so called hypothesis H.