A survey on the hypertranscendence of the solutions of the Schr\"oder's, B\"ottcher's and Abel's equations

TL;DR

This survey explores the historical and mathematical context of solutions to Schr"oder's, B"ottcher's, and Abel's equations, focusing on their hypertranscendence and differential algebraic properties as established by Becker and Bergweiler.

Contribution

It provides a comprehensive overview of the mathematical domains involved and clarifies how these areas contribute to understanding the solutions' hypertranscendence.

Findings

Solutions are differentially algebraic as classified by Becker and Bergweiler.

The equations connect various mathematical fields like iteration theory and differential algebra.

Historical development enriches understanding of the solutions' properties.

Abstract

In 1994, P.-G. Becker and W. Bergweiler listed all the differentially algebraic solutions of three famous functional equations: the Schr{\"o}der's, B{\"o}ttcher's and Abel's equations. The proof of this theorem combines various domains of mathematics. This goes from the theory of iteration, which gave birth to these equations, to the algebro-differential notion of coherent families developed by M. Boshernitzan and L. A. Rubel. This survey is an excursion into the history of these equations, in order to enlighten the different pieces of mathematics they bring together and how these parts fit into the result of P.-G. Becker and W. Bergweiler.