Curves homogeneous under analytic transformations

Abstract

We call a subset $K$ of $\mathbb{C}$ \emph{biholomorphically homogeneous} if for any two points $p,q\in K$ there exists a neighborhood $U$ of $p$ and a biholomorphism $\psi :U\to \psi (U)\subset \mathbb{C}$ such that $\psi (p)=q$ and $\psi (K\cap U)=K\cap \psi (U)$. We show that a biholomorphically homogeneous smooth curve $\gamma \subset \mathbb{C}$ is necessarily real-analytic. Furthermore we show that the same holds for the homogeneity with respect of a wider class of groups $G$ of real-analytic transformations of the plane. The result also extends to subsets $K\subset {\mathbb{R}}^2$ which are just locally closed.