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

Jaksztas, Ludwik

doi:10.1090/S0002-9947-2011-05208-6

On the derivative of the Hausdorff dimension of the quadratic Julia sets
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by Ludwik Jaksztas

Trans. Amer. Math. Soc. 363 (2011), 5251-5291

DOI: https://doi.org/10.1090/S0002-9947-2011-05208-6

Published electronically: April 21, 2011

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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 $c\nearrow 1/4$. 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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