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Cantor sets of circles of Sierpiński curve Julia sets

Published online by Cambridge University Press:  01 October 2007

ROBERT L. DEVANEY
ROBERT L. DEVANEY*
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (email: bob@bu.edu)
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Abstract

Our goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of parameters for which the corresponding Julia set is a Sierpiński curve. Hence, the Julia sets for each of these parameters are homeomorphic. However, each of the maps in this set is dynamically distinct from (i.e. not topologically conjugate to) any other map in this set (with only finitely many exceptions). We also show that, in the dynamical plane for any map drawn from a large open set in the connectedness locus in this family, there is a Cantor set of invariant simple closed curves on which the map is conjugate to the product of certain subshifts of finite type with the maps on the unit circle.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1525 - 1539
Copyright
Copyright © Cambridge University Press 2007

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