A sharp version of Shimizu's theorem on entire automorphic functions

TL;DR

This paper advances the understanding of automorphic groups of entire functions by proving new properties of their orbits, derivatives at fixed points, and characterizations of automorphic functions, culminating in an algorithm for their computation.

Contribution

It provides a sharp refinement of Shimizu's theorem, including new results on automorphic orbits, derivatives, and explicit forms, along with an algorithm for computing automorphic functions.

Findings

Automorphic group orbits have no finite accumulation points.

Derivative of automorphic functions at fixed points can be accurately computed.

Automorphic functions uniform over open subsets have a precise form.

Abstract

This paper develops further the theory of the automorphic group of non-constant entire functions. This theory essentially started with two remarkable papers of Tatsujir\^o Shimizu that were published in 1931. There are three results in this paper. The first result is that the ${\rm Aut}(f)$-orbit of any complex number has no finite accumulation point. The second result is an accurate computation of the derivative of an automorphic function of an entire function at any of its fixed points. The third result gives the precise form of an automorphic function that is uniform over an open subset of $\mathbb{C}$. This last result is a follow up of a remarkable theorem of Shimizu. It is a sharp form of his result. It leads to an algorithm of computing the entire automorphic functions of entire functions. The complexity is computed using an height estimate of a rational parameter discovered by Shimizu.