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Accumulation theorems for quadratic polynomials

Published online by Cambridge University Press:  19 September 2008

Dan Erik Krarup Sørensen
Dan Erik Krarup Sørensen
Affiliation:
Matematisk Institut, Bygn. 303, Danmarks Tekniske Universitet, DK-2800 Lyngby, Danmark (e-mail: dan@mat.dtu.dk)
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Abstract

We consider the one-parameter family of quadratic polynomials:

i.e. monic centered quadratic polynomials with an indifferent fixed point αt and prefixed point −αt. Let At, be any one of the sets {0, ±αt}, {±αt}, {0, αt}, or {0, −αt}. Then we prove that for quadratic Julia sets corresponding to a Gδ-dense subset of there is an explicitly given external ray accumulating on At. In the case At = {±αt} the theorem is known as the Douady accumulation theorem.

Corollaries are the non-local connectivity of these Julia sets and the fact that all such Julia sets contain a Cremer point. Existence of non-locally connected quadratic Julia sets of Hausdorff dimension two is derived by using a recent result of Shishikura. By tuning, the results hold on the boundary of any hyperbolic component of the Mandelbrot set.

Finally, we concentrate on quadratic Cremer point polynomials. Here we prove that any ray accumulating on two symmetrical points of the Julia set must accumulate the origin. As a consequence, the dense Gδsets arising from the first two possible choices of At are the same. We also prove that, if two distinct rays accumulate both to two distinct points, then the rays must accumulate on a common continuum joining the two points. This supports the conjecture that αt and –αt may be joined by an arc in the Julia set.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 3 , June 1996 , pp. 555 - 590
Copyright
Copyright © Cambridge University Press 1996

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