Annales Academi� Scientiarum Fennic�
Mathematica
Volumen 32, 2007, 83-98

APOLLONIAN ISOMETRIES OF REGULAR DOMAINS ARE M�BIUS MAPPINGS

Peter H�st� and Zair Ibragimov

University of Oulu, Department of Mathematical Sciences
P.O. Box 3000, FI-90014 University of Oulu, Finland; peter.hasto 'at' helsinki.fi

University of Cincinnati, Department of Mathematical Sciences
P.O. Box 210025, Cincinnati, OH 45221-0025, U.S.A.; ibragizs 'at' math.uc.edu

Abstract. The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. However, since it was introduced by Beardon in 1998, it has remained an open question what its isometries are. Beardon first raised this question and asked if the Apollonian isometries were just M�bius mappings. In this paper we show that this is the case in open sets with regular, for instance C¹, boundary.

2000 Mathematics Subject Classification: Primary 30F45; Secondary 30C65.

Key words: Apollonian metric, Barbilian metric, isometries, M�bius mappings.

Reference to this article: P. H�st� and Z. Ibragimov: Apollonian isometries of regular domains are M�bius mappings. Ann. Acad. Sci. Fenn. Math. 32 (2007), 83-98.

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