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Symbolic dynamics for angle-doubling on the circle III. Sturmian sequences and the quadratic map

Published online by Cambridge University Press:  19 September 2008

Karsten Keller
Karsten Keller
Affiliation:
Fachbereich Mathematik, Ernst-Moritz-Arndt-Universität, D-O-2200 Greifswald, Germany
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Abstract

By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling map h on the circle T. In particular, a connected and locally connected Julia set can be considered as a topological factor T/ ≈ of T with respect to a special h -invariant equivalence relation ≈ on T, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α → α from T onto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point α, they gave a topological classification of the abstract Julia sets T/ α. It turns out that T/ α contains simple closed curves iff the point α has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 4 , December 1994 , pp. 787 - 805
Copyright
Copyright © Cambridge University Press 1994

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