Exponential automorphisms and a problem of Mycielski

TL;DR

This paper investigates exponential automorphisms of the complex numbers, providing partial answers to Mycielski's questions about their fixed points, specifically regarding logarithms and roots of numbers.

Contribution

It characterizes the fixed points of exponential automorphisms of , answering Mycielski's questions up to roots of unity and multiples of 27i.

Findings

Answers are modulo a multiple of 27i and roots of unity.

Provides conditions under which  automorphisms fix certain logarithms and roots.

Clarifies the structure of exponential automorphisms in relation to Mycielski's problem.

Abstract

An exponential automorphism of $\mathbf{C}$ is a function $\alpha: \mathbf{C} \rightarrow \mathbf{C}$ such that $\alpha(z_1 + z_2) = \alpha(z_1) + \alpha(z_2)$ and $\alpha\left( e^z \right) = e^{\alpha(z)}$ for all $z, z_1, z_2 \in \mathbf{C}$. Jan Mycielski asked if $\alpha(\ln 2) = \ln 2$ and if $\alpha(2^{1/k}) = 2^{1/k}$ for $k = 2, 3, 4$ and for all exponential automorphisms $\alpha$. These questions are answered modulo a multiple of $2\pi i$ and a root of unity.