Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 32, 2007, 83-98

# APOLLONIAN ISOMETRIES OF REGULAR
DOMAINS ARE MÖBIUS MAPPINGS

## Peter Hästö 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 Möbius mappings.
In this paper we show that this is the case in open sets with regular,
for instance *C*^{1}, boundary.

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

**Key words:**
Apollonian metric, Barbilian metric, isometries, Möbius mappings.

**Reference to this article:** P. Hästö and Z. Ibragimov:
Apollonian isometries of regular domains are Möbius mappings.
Ann. Acad. Sci. Fenn. Math. 32 (2007), 83-98.

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Copyright © 2007 by Academia Scientiarum Fennica