Dynamics in the Eremenko-Lyubich class

TL;DR

This paper surveys the dynamics of transcendental entire functions in the Eremenko-Lyubich class, highlighting their similarities to polynomial dynamics and summarizing key results, techniques, and open questions in the field.

Contribution

It provides a comprehensive overview of the well-developed theory of class  functions, emphasizing recent advances and foundational concepts for future research.

Findings

Class  functions exhibit polynomial-like dynamics.

Many powerful tools have been developed for analyzing these functions.

Understanding transcendental dynamics can inform polynomial and rational dynamics.

Abstract

The study of the dynamics of polynomials is now a major field of research, with many important and elegant results. The study of entire functions that are not polynomials -- in other words transcendental entire functions -- is somewhat less advanced, in part due to certain technical differences compared to polynomial or rational dynamics. In this paper we survey the dynamics of functions in the Eremenko-Lyubich class, $\mathcal{B}$. Among transcendental entire functions, those in this class have properties that make their dynamics most "similar" to the dynamics of a polynomial, and so particularly accessible to detailed study. Many authors have worked in this field, and the dynamics of class $\mathcal{B}$ functions is now particularly well-understood and well-developed. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. There is also an increasing expectation that learning new results in transcendental dynamics will lead to a better understanding of the polynomial and rational cases. We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give stepping-stones to deeper inquiry.