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arXiv · 2209.01027

Exponential automorphisms and a problem of Mycielski

Melvyn B. Nathanson
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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.

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Melvyn B. Nathanson. 2022-09-02. Exponential automorphisms and a problem of Mycielski. https://arxiv.org/abs/2209.01027

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