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

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- by Mariusz Urbanski and Michel Zinsmeister
- Conform. Geom. Dyn.
**5**(2001), 140-152 - DOI: https://doi.org/10.1090/S1088-4173-01-00070-4
- Published electronically: October 18, 2001
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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 $J(f_{0},g_{\sigma })$.## References

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## Bibliographic 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