Valiron, Nevanlinna and Picard exceptional sets of iterations of rational functions

Abstract

For every rational function of degree more than one, there exists a transcendental meromorphic solution of the Schröder equation. By Yanagihara and Eremenko-Sodin, it is known that the Valiron, Nevanlinna and Picard exceptional sets of this solution are all same.

As an analogue of this result, we show that all the Valiron, Nevanlinna and Picard exceptional sets of iterations of a rational function of degree more than one are also same. As a corollary, the equidistribution theorem in complex dynamics follows.