Article

The Journal of Geometric Analysis

, 10:169

First online:

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

  • Dan Erik Krarup SørensenAffiliated with

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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

58F08 30D05

Key Words and Phrases

quadratic polynomial Julia set Mandelbrot set local connectivity accumulation set prime end impression external ray tuning satellite wake infinitely renormalizable hyperbolic components robustness