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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Continuity of Hausdorff dimension of Julia-Lavaurs sets as a function of the phase
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by Mariusz Urbanski and Michel Zinsmeister PDF
Conform. Geom. Dyn. 5 (2001), 140-152 Request permission

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 $J(f_{0},g_{\sigma })$.
References
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Additional Information
  • Mariusz Urbanski
  • Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
  • Email: urbanski@unt.edu
  • Michel Zinsmeister
  • Affiliation: Mathématiques, Université d’Orleans, BP 6759 45067 Orléans Cedex, France
  • Email: Michel.Zinsmeister@labomath.univ-orleans.fr
  • 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 5 (2001), 140-152
  • MSC (2000): Primary 37F45; Secondary 37F35, 37F15
  • DOI: https://doi.org/10.1090/S1088-4173-01-00070-4
  • MathSciNet review: 1872160