Thermodynamic Formalism and Variations of the Hausdorff Dimension of Quadratic Julia Sets

Let d(c) denote the Hausdorff dimension of the Julia set of the polynomial z? z²+czmapsto z^2+c. The function d restricted to [0,+X) is real analytic in [0,frac14)è(frac14,+￥)[0,frac{1}{4})cup (frac{1}{4},+infty) ([Ru²]), is left-continuous at ¼ ([Bo,Zi]) but not continuous ([Do,Se,Zi]). We prove that c? d￠(c)cmapsto d'(c) tends to + X from the left at ¼ as (frac14-c)^{d(frac14)-frac32}(frac{1}{4}-c)^{d(frac{1}{4})-frac{3}{2}}. In particular the graph of d has a vertical tangent on the left at ¼, a result which supports the numerical experiments.