Exponential iteration and Borel sets

TL;DR

This paper classifies the Borel complexity of points escaping to infinity under exponential iteration and shows topological equivalences among non-escaping Julia sets for these functions.

Contribution

It precisely determines the Borel class of escaping points and establishes topological equivalences among their Julia sets, advancing understanding of exponential dynamics.

Findings

Exact Borel class of escaping points identified

Topological equivalence of non-escaping Julia sets proven

Provides new insights into exponential iteration dynamics

Abstract

We determine the exact Borel class of the points whose iterates under $\exp(z)+a$ tend to infinity. We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.