Tame sets, dominating maps, and complex tori

Author:
Gregery T. Buzzard

Journal:
Trans. Amer. Math. Soc. **355** (2003), 2557-2568

MSC (2000):
Primary 32H02; Secondary 32E30

Published electronically:
December 18, 2002

MathSciNet review:
1974003

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Abstract: A discrete subset of is said to be tame if there is an automorphism of taking the given discrete subset to a subset of a complex line; such tame sets are known to allow interpolation by automorphisms. We give here a fairly general sufficient condition for a discrete set to be tame. In a related direction, we show that for certain discrete sets in there is an injective holomorphic map from into itself whose image avoids an -neighborhood of the discrete set. Among other things, this is used to show that, given any complex -torus and any finite set in this torus, there exist an open set containing the finite set and a locally biholomorphic map from into the complement of this open set.

**1.**Eric Bedford and Victoria Pambuccian,*Dynamics of shift-like polynomial diffeomorphisms of 𝐶^{𝑁}*, Conform. Geom. Dyn.**2**(1998), 45–55 (electronic). MR**1624646**, 10.1090/S1088-4173-98-00027-7**2.**Gregery T. Buzzard and Franc Forstneric,*An interpolation theorem for holomorphic automorphisms of 𝐶ⁿ*, J. Geom. Anal.**10**(2000), no. 1, 101–108. MR**1758584**, 10.1007/BF02921807**3.**Gregery T. Buzzard and John H. Hubbard,*A Fatou-Bieberbach domain avoiding a neighborhood of a variety of codimension 2*, Math. Ann.**316**(2000), no. 4, 699–702. MR**1758449**, 10.1007/s002080050350**4.**Stephen J. Gardiner,*Harmonic approximation*, London Mathematical Society Lecture Note Series, vol. 221, Cambridge University Press, Cambridge, 1995. MR**1342298****5.**Mark L. Green,*Holomorphic maps to complex tori*, Amer. J. Math.**100**(1978), no. 3, 615–620. MR**501228**, 10.2307/2373842**6.**Jean-Pierre Rosay and Walter Rudin,*Holomorphic maps from 𝐶ⁿ to 𝐶ⁿ*, Trans. Amer. Math. Soc.**310**(1988), no. 1, 47–86. MR**929658**, 10.1090/S0002-9947-1988-0929658-4**7.**Jean-Pierre Rosay and Walter Rudin,*Arakelian’s approximation theorem*, Amer. Math. Monthly**96**(1989), no. 5, 432–434. MR**994035**, 10.2307/2325151**8.**Jean-Pierre Rosay and Walter Rudin,*Growth of volume in Fatou-Bieberbach regions*, Publ. Res. Inst. Math. Sci.**29**(1993), no. 1, 161–166. MR**1208033**, 10.2977/prims/1195167547

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Additional Information

**Gregery T. Buzzard**

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

Email:
buzzard@math.purdue.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03229-4

Received by editor(s):
June 20, 1999

Published electronically:
December 18, 2002

Additional Notes:
Supported in part by an NSF Postdoctoral Fellowship

Article copyright:
© Copyright 2002
American Mathematical Society