A landing theorem for dynamic rays of geometrically finite entire functions

TL;DR

This paper proves that geometrically finite entire functions of finite order have dynamic rays connecting repelling or parabolic points to infinity, extending the understanding of their escaping sets and dynamic structure.

Contribution

It establishes the existence of dynamic rays for geometrically finite entire functions of finite order, even under weaker conditions than finite order.

Findings

Existence of dynamic rays connecting repelling/parabolic points to infinity.

Dynamic rays exist for functions with finite order or finite composition of finite-order functions.

Provides a new understanding of the escaping set structure for these functions.

Abstract

A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective curve consisting of escaping points of f that connects z_0 to infinity. (This curve is called a dynamic ray.) In fact, the assumption of finite order can be weakened considerably; for example, it is sufficient to assume that f can be written as a finite composition of finite-order functions.