Equivalence of domains with isomorphic semigroups of endomorphisms
HTML articles powered by AMS MathViewer
- by Sergei Merenkov PDF
- Proc. Amer. Math. Soc. 130 (2002), 1743-1753 Request permission
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 $\mathbb {C}^n$.References
- V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York, 1988. Translated from the Russian by Joseph Szücs [József M. Szűcs]. MR 947141, DOI 10.1007/978-1-4612-1037-5
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
- A. Erëmenko, On the characterization of a Riemann surface by its semigroup of endomorphisms, Trans. Amer. Math. Soc. 338 (1993), no. 1, 123–131. MR 1106188, DOI 10.1090/S0002-9947-1993-1106188-2
- Maurice Heins, Complex function theory, Pure and Applied Mathematics, Vol. 28, Academic Press, New York-London, 1968. MR 0239054
- A. Hinkkanen, Functions conjugating entire functions to entire functions and semigroups of analytic endomorphisms, Complex Variables Theory Appl. 18 (1992), no. 3-4, 149–154. MR 1157923, DOI 10.1080/17476939208814541
- Kenneth Hoffman and Ray Kunze, Linear algebra, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0276251
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- Hej Iss’sa, On the meromorphic function field of a Stein variety, Ann. of Math. (2) 83 (1966), 34–46. MR 185143, DOI 10.2307/1970468
- Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- K. D. Magill Jr., A survey of semigroups of continuous selfmaps, Semigroup Forum 11 (1975/76), no. 3, 189–282. MR 393330, DOI 10.1007/BF02195270
Additional Information
- Sergei Merenkov
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: smerenko@math.purdue.edu
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1743-1753
- MSC (2000): Primary 32A10, 08A35
- DOI: https://doi.org/10.1090/S0002-9939-01-06409-7
- MathSciNet review: 1887022