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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the derivative of the Hausdorff dimension of the quadratic Julia sets


Author: Ludwik Jaksztas
Journal: Trans. Amer. Math. Soc. 363 (2011), 5251-5291
MSC (2000): Primary 37F45; Secondary 37F35
Published electronically: April 21, 2011
MathSciNet review: 2813415
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Abstract: Let $ d(c)$ denote the Hausdorff dimension of the Julia set $ J_c$ of the polynomial $ f_c(z)=z^2+c$. The function $ c\mapsto d(c)$ is real-analytic on the interval $ (-3/4,1/4)$, which is included in the main cardioid of the Mandelbrot set. It was shown by G. Havard and M. Zinsmeister that the derivative $ d'(c)$ tends to $ +\infty$ as fast as $ (1/4-c)^{d(1/4)-3/2}$ when $ c\nearrow1/4$. Under numerically verified assumption $ d(-3/4)<4/3$, we prove that $ d'(c)$ tends to $ -\infty$ as $ -(c+3/4)^{3d(-3/4)/2-2}$ when $ c\searrow-3/4$.


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Additional Information

Ludwik Jaksztas
Affiliation: Faculty of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland
Email: jaksztas@impan.gov.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05208-6
PII: S 0002-9947(2011)05208-6
Received by editor(s): October 19, 2008
Received by editor(s) in revised form: July 11, 2009
Published electronically: April 21, 2011
Additional Notes: This work was partially supported by Polish MNiSW grants 2P03A03425, NN201 0222 33, and EU FP6 Marie Curie RTN CODY at Orléans France.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.