Formalisme thermodynamique et variations de la dimension de Hausdorff des ensembles de Julia des polynômes quadratiques
Résumé
Let d(c) denote the Hausdorff dimension of the Julia set of the polynomial $z\mapsto z^2+c$. The function d restricted to [0,+X) is real analytic in $[0,\frac{1}{4})\cup (\frac{1}{4},+\infty)$ ([Ru2]), is left-continuous at ¼ ([Bo,Zi]) but not continuous ([Do,Se,Zi]). We prove that $c\mapsto d'(c)$ tends to + X from the left at ¼ as $(\frac{1}{4}-c)^{d(\frac{1}{4})-\frac{3}{2}}$. In particular the graph of d has a vertical tangent on the left at ¼, a result which supports the numerical experiments.