Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We show that the Julia set of a non-elementary rational semigroup $G$is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of $G$. 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.


References [Enhancements On Off] (What's this?)

  • 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.

Similar Articles

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

American Mathematical Society