Singularities of Schr\"oder maps and unhyperbolicity of rational functions

Abstract

We study transcendental singularities of a Schr\"oder map arising from a rational function $f$, using results from complex dynamics and Nevanlinna theory. These maps are transcendental meromorphic functions of finite order in the complex plane. We show that their transcendental singularities lie over the set where $f$ is not semihyperbolic (unhyperbolic). In addition, if they are direct, then they lie over only attracting periodic points of $f$, and moreover, if $f$ is a polynomial, then both direct and indirect singularities lie over attracting, parabolic and Cremer periodic points of $f$. We also obtain concrete examples of both kinds of transcendental singularities of Schr\"oder maps as well as a new proof of the Pommerenke-Levin-Yoccoz inequality and a new formulation of the Fatou conjecture.Comment: 17 pages; some typos correcte