Equivalence of domains with isomorphic semigroups of endomorphisms

Abstract:

For two bounded domains $\Omega _1,\ \Omega _2$ in $\mathbb {C}$ whose semigroups of analytic endomorphisms $E(\Omega _1), \ E(\Omega _2)$ are isomorphic with an isomorphism $\varphi :\ E(\Omega _1)\rightarrow E(\Omega _2)$, Eremenko proved in 1993 that there exists a conformal or anticonformal map $\psi :\ \Omega _1\rightarrow \Omega _2$ such that $\varphi f=\psi \circ f\circ \psi ^{-1},$ for all $f\in E(\Omega _1)$. In the present paper we prove an analogue of this result for the case of bounded domains in $\mathbb {C}^n$.