Exponential automorphisms and a problem of Mycielski

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.