Equivalence of domains with isomorphic semigroups of endomorphisms

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 $\mathbb{C} ^n$.