Equivalence of domains with isomorphic semigroups of endomorphisms

TL;DR

This paper extends a 1993 result by Eremenko, demonstrating that for bounded domains in complex n-space, isomorphic semigroups of analytic endomorphisms imply the domains are equivalent via a conformal or anticonformal map.

Contribution

It generalizes Eremenko's 1993 theorem from the complex plane to higher-dimensional complex spaces, establishing domain equivalence through semigroup isomorphisms.

Findings

Semigroups of endomorphisms determine domain equivalence

Isomorphism implies conjugation by conformal or anticonformal maps

Generalization from  to n in complex analysis

Abstract

For two bounded domains in the complex plane whose semigroups of analytic endomorphisms are isomorphic, Eremenko proved in 1993 that the isomorphism is given as a conjugation by a conformal or anticonformal map. In the present paper we prove an analogue of this result for the case of bounded domains in \mathbb C^n.