Abstract
Let d ( c ) d(c) denote the Hausdorff dimension of the Julia set J c J_c of the polynomial f c ( z ) = z 2 + c f_c(z)=z^2+c . The function c ↦ d ( c ) c\mapsto d(c) is real-analytic on the interval ( − 3 / 4 , 1 / 4 ) (-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 ) d’(c) tends to + ∞ +\infty as fast as ( 1 / 4 − c ) d ( 1 / 4 ) − 3 / 2 (1/4-c)^{d(1/4)-3/2} when c ↗ 1 / 4 c\nearrow 1/4 . Under numerically verified assumption d ( − 3 / 4 ) > 4 / 3 d(-3/4)>4/3 , we prove that d ′ ( c ) d’(c) tends to − ∞ -\infty as − ( c + 3 / 4 ) 3 d ( − 3 / 4 ) / 2 − 2 -(c+3/4)^{3d(-3/4)/2-2} when c ↘ − 3 / 4 c\searrow -3/4 .
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