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.”
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Sørensen, D.E.K. Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set. J Geom Anal 10, 169–206 (2000). https://doi.org/10.1007/BF02921810
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DOI: https://doi.org/10.1007/BF02921810