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Equivalence of domains with isomorphic semigroups of endomorphisms

Authors:	Sergei Merenkov

Institution:	Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

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 $\mathbb{C} ^n$ .

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