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Conformal Geometry and Dynamics

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The dual nest for degenerate Yoccoz puzzles

Author: Magnus Aspenberg
Journal: Conform. Geom. Dyn. 13 (2009), 187-196
MSC (2000): Primary 37F20; Secondary 30D05
Published electronically: July 27, 2009
MathSciNet review: 2525102
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Abstract: The Yoccoz puzzle is a fundamental tool in Holomorphic Dynamics. The original combinatorial argument by Yoccoz, based on the Branner-Hubbard tableau, counts the preimages of a non-degenerate annulus in the puzzle. However, in some important new applications of the puzzle (notably, matings of quadratic polynomials) there is no non-degenerate annulus. We develop a general combinatorial argument to handle this situation. It allows us to derive corollaries, such as the local connectedness of the Julia set, for suitable families of rational maps.

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Additional Information

Magnus Aspenberg
Affiliation: Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Ludewig-Meyn Str.4, 24 098 Kiel, Germany

Received by editor(s): April 8, 2009
Published electronically: July 27, 2009
Additional Notes: The author gratefully acknowledges funding from the Research Training Network CODY of the European Commission
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.