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)

 
 

 

Not all Julia sets are quasi-self-similar


Author: Pentti Järvi
Journal: Proc. Amer. Math. Soc. 125 (1997), 835-837
MSC (1991): Primary 30D05, 58F08
DOI: https://doi.org/10.1090/S0002-9939-97-03706-4
MathSciNet review: 1363426
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that there exist rational functions, whose Julia set fails to be quasi-self-similar.


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

  • 1. P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), 85-141. MR 85h:58001
  • 2. L. Carleson and T. W. Gamelin, Complex Dynamics, Springer-Verlag, Berlin, 1993. MR 94h:30033
  • 3. L. Keen, Julia sets, Chaos and Fractals: The Mathematics Behind the Computer Graphics (L. Keen and R. Devaney, eds.), Amer. Math. Soc., Providence, 1989, pp. 57-74. MR 91a:58130
  • 4. J. McLaughlin, A note on Hausdorff measures of quasi-self-similar sets, Proc. Amer. Math. Soc. 100 (1987), 183-186. MR 88d:54054
  • 5. M. Shishikura, The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets, Preprint 1991/7, SUNY Stony Brook, IMS.
  • 6. M. Shishikura, The boundary of the Mandelbrot set has Hausdorff dimension two, Astérisque 222 (1994), 389-405. CMP 94:15
  • 7. D. Sullivan, Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorff dimension two, Geometry Symposium Utrecht 1980 (E. Looijenga, D. Siersma, and F. Takens, eds.), Lecture Notes in Math., vol. 894, Springer-Verlag, Berlin, 1981, pp. 127-144. MR 83h:53054
  • 8. D. Sullivan, Conformal dynamical systems, Geometric Dynamics (J. Palis, ed.), Lecture Notes in Math., vol. 1007, Springer-Verlag, Berlin, 1983, pp. 725-752. MR 85m:58112
  • 9. P. Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math. 152 (1984), 127-140. MR 85m:30031
  • 10. J. Väisälä, Porous sets and quasisymmetric maps, Trans. Amer. Math. Soc. 299 (1987), 525-533. MR 88a:30049

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D05, 58F08

Retrieve articles in all journals with MSC (1991): 30D05, 58F08


Additional Information

Pentti Järvi
Affiliation: Department of Mathematics, University of Helsinki, P.O. Box 4 (Hallituskatu 15), FIN-00014 Helsinki, Finland

DOI: https://doi.org/10.1090/S0002-9939-97-03706-4
Keywords: Iteration, rational function, Julia set, quasi-self-similar set, porous set.
Received by editor(s): September 19, 1995
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society