Exponential automorphisms and a problem of Mycielski

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

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