Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's

Author:
Rich Stankewitz

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2569-2575

MSC (1991):
Primary 30D05, 58F23

DOI:
https://doi.org/10.1090/S0002-9939-00-05313-2

Published electronically:
February 29, 2000

MathSciNet review:
1662218

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We show that the Julia set of a non-elementary rational semigroup is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of . This also proves that the limit set of a non-elementary Möbius group is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of the group and this implies that the limit set of a finitely generated non-elementary Kleinian group is uniformly perfect.

**1.**I. N. Baker.

Repulsive fixed points of entire functions.*Math. Z.*, 104:252-256, 1968. MR**37:1599****2.**A. F. Beardon and Ch. Pommerenke.

The Poincare metric of plane domains.*J. London Math. Soc.*, 18:475-483, 1978. MR**80a:30020****3.**Alan F. Beardon.*The geometry of discrete groups*.

Springer-Verlag, New York, 1983. MR**85d:22026****4.**Alan F. Beardon.*Iterations of Rational Functions*.

Springer-Verlag, New York, 1991. MR**92j:30026****5.**David Boyd.

An invariant measure for finitely generated rational semigroups.*Complex Variables*, to appear.**6.**Lennart Carleson and Theodore W. Gamelin.*Complex Dynamics*.

Springer-Verlag, New York, 1993. MR**94h:30033****7.**A. Eremenko.

Julia sets are uniformly perfect.*Preprint, Purdue University*, 1992.**8.**Kenneth Falconer.*Fractal Geometry, Mathematical foundations and Applications*.

John Wiley and Sons, 1990. MR**92j:28008****9.**A. Hinkkanen.

Julia sets of rational functions are uniformly perfect.*Math. Proc. Camb. Phil.*, 113:543-559, 1993. MR**94b:58084****10.**A. Hinkkanen and G.J. Martin.

The dynamics of semigroups of rational functions I.*Proc. London Math. Soc.(3)*, 73:358-384, 1996. MR**97e:58198****11.**A. Hinkkanen and G.J. Martin.

Julia sets of rational semigroups.*Math. Z.*, 222(2):161-169, 1996. MR**98d:30038****12.**John E. Hutchinson.

Fractals and self similarity.*Indiana University Math. Journal*, 30:731-747, 1981. MR**82h:49026****13.**R. Mañé and L. F. da Rocha.

Julia sets are uniformly perfect.*Proc. Amer. Math. Soc.*, 116:251-257, 1992. MR**92k:58229****14.**Ch. Pommerenke.

Uniformly perfect sets and the poincare metric.*Arch. Math.*, 32:192-199, 1979. MR**80j:30073****15.**Ch. Pommerenke.

On uniformly perfect sets and Fuchsian groups.*Analysis*, 4:299-321, 1984. MR**86e:30044****16.**Fu-Yao Ren.

Advances and problems in random dynamical systems.

Preprint, 1998.**17.**Wilhelm Schwick.

Repelling periodic points in the Julia set.*Bull. London Math. Soc.*, 29:314-316, 1997. MR**97m:30029****18.**Rich Stankewitz.

Completely invariant sets of normality for rational semigroups.*Complex Variables*.

to appear.**19.**Rich Stankewitz.*Completely invariant Julia sets of rational semigroups*.

PhD thesis, University of Illinois, 1998.**20.**Rich Stankewitz.

Completely invariant Julia sets of polynomial semigroups.*Proc. Amer. Math. Soc.*, 127:2889-2898, 1999. CMP**98:07****21.**Hiroki Sumi.

On hausdorff dimension of Julia sets of hyperbolic rational semigroups.*Kodai Math. J.*, 21:10-28, 1998. CMP**98:12****22.**Hiroki Sumi.

On dynamics of hyperbolic rational semigroups.*Journal of Mathematics of Kyoto University*, to appear.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
30D05,
58F23

Retrieve articles in all journals with MSC (1991): 30D05, 58F23

Additional Information

**Rich Stankewitz**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

Email:
richs@math.tamu.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05313-2

Keywords:
Rational semigroups,
Kleinian groups,
Julia sets,
uniformly perfect,
iterated function systems

Received by editor(s):
August 26, 1998

Received by editor(s) in revised form:
October 5, 1998

Published electronically:
February 29, 2000

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 2000
American Mathematical Society