The Journal of Geometric Analysis

, 10:169

Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set

  • Dan Erik Krarup Sørensen
Article

DOI: 10.1007/BF02921810

Cite this article as:
Sørensen, D.E.K. J Geom Anal (2000) 10: 169. doi:10.1007/BF02921810

Abstract

We present two strategies for producing and describing some connected non-locally connected Julia sets of infinitely renormalizable quadratic polynomials. By using a more general strategy, we prove that all of these Julia sets fail to be arc-wise connected, and that their critical point is non-accessible.

Using the first strategy we prove the existence of polynomials having an explicitly given external ray accumulating two particular, symmetric points. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists spiral.”

A weaker result is obtained using the second strategy: the existence of polynomials having an explicitly given external ray accumulating at a particular point, but having in its impression the symmetric point as well. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists sine.”

Math Subject Classification

58F0830D05

Key Words and Phrases

quadratic polynomialJulia setMandelbrot setlocal connectivityaccumulation setprime endimpressionexternal raytuningsatellitewakeinfinitely renormalizablehyperbolic componentsrobustness

Copyright information

© Mathematica Josephina, Inc. 2000

Authors and Affiliations

  • Dan Erik Krarup Sørensen
    • 1
  1. 1.Høje TaastrupDenmark