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Equivalence of domains with isomorphic semigroups of endomorphisms

Author: Sergei Merenkov
Journal: Proc. Amer. Math. Soc. 130 (2002), 1743-1753
MSC (2000): Primary 32A10, 08A35
Published electronically: November 9, 2001
MathSciNet review: 1887022
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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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Additional Information

Sergei Merenkov
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Received by editor(s): December 12, 2000
Published electronically: November 9, 2001
Additional Notes: This research was supported by NSF, DMS 0072197
Communicated by: Steven R. Bell
Article copyright: © Copyright 2001 American Mathematical Society

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