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Quadratic Julia sets and irrationally indifferent periodic orbits
by
John. C. Mayer
University of Alabama- Birmingham
When a quadratic map f:z --> z2+c has an irrationally indifferent periodic orbit O, the topological structure of the corresponding Julia set J=J(f) is not yet well understood, though recent progress has been made by Kiwi, Rogers, Perez-Marco, Petersen, Shishikura, Sorensen, and the authors. (Such cases correspond to parameter values c on the boundary of hyperbolic components of the interior of the Mandelbrot set.) It is known that the periodic orbit O ``attracts, " in an appropriate sense, at least one critical point. We show that J contains a periodic subcontinuum B, which we call a ``building block, " whose orbit contains the orbit of the critical point and either O or the boundary of a Siegel disk about O. We discuss the topological structure of the resulting Julia set J and how this structure is related to the induced map on the circle at infinity of external rays.
This is the result of joint work with Lex G. Oversteegen and Joachim Grispolakis, and conversations with Alexander M. Blokh.
Date received: July 8, 1997
Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-90.