Escaping Set of e^z-1

TL;DR

This paper studies the escaping set of the complex function f(z) = e^z - 1, revealing its structure as a union of rays and infinite sets where points tend to infinity under iteration.

Contribution

It characterizes the detailed topological structure of the escaping set for f(z) = e^z - 1, including its decomposition into rays and infinite sets.

Findings

The escaping set is a disjoint union of countably many rays.

It also forms an uncountable union of infinite sets.

Points in these sets escape to infinity through specific pathways.

Abstract

We investigate the set I(f) of points that converge to infinity under iteration of the map f(z) = e^z-1 and show that it is the disjoint union of countably many rays and uncountable union of infinite sets whose points escape to infinity through the sets.