# On the Boundary Behaviour of Polymorphic Functions

On the Boundary Behaviour of Polymorphic Functions Let Φ be a Fuchsian group acting on the unit disk D. This group may be infinitely generated and may have elliptic elements. A non-constant function f meromorphic in D is called Φ-polymorphic if, for every ϕ ∈Φ there is a Moebius transformation γ such that f ∘ φ = γ ∘ f. This induces a homomorphism f* of Φ into PSL(2, ℂ). The image group Г:= f *(Φ) need not be discrete. In this paper we study function-theoretic properties of a polymorphic function f and its relation to the limit sets and fixed points of Φ and the image group Г. This has some consequences for degenerate Kleinian groups. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# On the Boundary Behaviour of Polymorphic Functions

, Volume 12 (1) – Jan 21, 2012
12 pages

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321822
Publisher site
See Article on Publisher Site

### Abstract

Let Φ be a Fuchsian group acting on the unit disk D. This group may be infinitely generated and may have elliptic elements. A non-constant function f meromorphic in D is called Φ-polymorphic if, for every ϕ ∈Φ there is a Moebius transformation γ such that f ∘ φ = γ ∘ f. This induces a homomorphism f* of Φ into PSL(2, ℂ). The image group Г:= f *(Φ) need not be discrete. In this paper we study function-theoretic properties of a polymorphic function f and its relation to the limit sets and fixed points of Φ and the image group Г. This has some consequences for degenerate Kleinian groups.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Jan 21, 2012

### References

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