TY - JOUR

T1 - Limit functions for convergence groups and uniformly quasiregular maps

AU - Hinkkanen, Aimo

AU - Martin, Gaven

N1 - Funding Information:
Research was partially supported by a grant from the Marsden Fund (NZ). This material is based on work supported by the National Science Foundation under grant Nos. 0200752 and 0457291.

PY - 2006/6

Y1 - 2006/6

N2 - We investigate the limit functions of iterates of a function belonging to a convergence group or of a uniformly quasiregular mapping. We show that it is not possible for a subsequence of iterates to tend to a non-constant limit function, and for another subsequence of iterates to tend to a constant limit function. It follows that the closure of the stabiliser of a Siegel domain for a uniformly quasiregular mapping is a compact abelian Lie group, which we further conjecture to be infinite. This result concerning possible limits of convergent subsequences of iterates for holomorphic rational functions on the Riemann sphere is known, and the only known method of proof involves universal covering surfaces and Möbius groups. Hence, our method yields a new and perhaps more elementary proof also in that case.

AB - We investigate the limit functions of iterates of a function belonging to a convergence group or of a uniformly quasiregular mapping. We show that it is not possible for a subsequence of iterates to tend to a non-constant limit function, and for another subsequence of iterates to tend to a constant limit function. It follows that the closure of the stabiliser of a Siegel domain for a uniformly quasiregular mapping is a compact abelian Lie group, which we further conjecture to be infinite. This result concerning possible limits of convergent subsequences of iterates for holomorphic rational functions on the Riemann sphere is known, and the only known method of proof involves universal covering surfaces and Möbius groups. Hence, our method yields a new and perhaps more elementary proof also in that case.

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U2 - 10.1112/S0024610706022691

DO - 10.1112/S0024610706022691

M3 - Article

AN - SCOPUS:33749334159

SN - 0024-6107

VL - 73

SP - 716

EP - 726

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 3

ER -