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Conformal Geometry and Dynamics

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Continuity of Hausdorff dimension of Julia-Lavaurs sets as a function of the phase

Authors: Mariusz Urbanski and Michel Zinsmeister
Journal: Conform. Geom. Dyn. 5 (2001), 140-152
MSC (2000): Primary 37F45; Secondary 37F35, 37F15
Published electronically: October 18, 2001
MathSciNet review: 1872160
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Abstract: Let $f_{0}(z)=z^{2}+1/4$ and ${\mathcal E}_{0}$ the set of phases $\overline {\sigma }$ such that the critical point $0$ escapes in one step by the Lavaurs map $g_{\sigma }$; it is a topological strip in the cylinder of phases whose boundary consists of two Jordan curves symmetric wrt $\mathbb R/ \mathbb Z$. We prove that if $\overline {\sigma }_{n}\in {\mathcal E}_{0}$ converges to $\overline {\sigma }\in \partial {\mathcal E}_{0}$ in such a way that $g_{\sigma _{n}}(0)$ converges to $g_{\sigma }(0)$ along an external ray, then the Hausdorff dimension of the Julia-Lavaurs set $J(f_{0}, g_{\sigma _{n}})$ converges to the Hausdorff dimension of $J(f_{0},g_{\sigma })$.

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Additional Information

Mariusz Urbanski
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430

Michel Zinsmeister
Affiliation: Mathématiques, Université d’Orleans, BP 6759 45067 Orléans Cedex, France

Received by editor(s): September 18, 2000
Received by editor(s) in revised form: June 28, 2001
Published electronically: October 18, 2001
Additional Notes: The research of the first author was partially supported by the NSF Grant DMS 9801583. He wishes to thank the University of Orleans and IHES, where a part of the research was done, for warm hospitality and excellent working conditions
Article copyright: © Copyright 2001 American Mathematical Society