Thermodynamic formalism and variations of the Hausdorff dimension of quadratic Julia sets (Q1976646)

This paper deals with variations of the Hausdorff dimension of quadratic Julia sets. Let \(P_c\) be a quadratic family of polynomials \[ P_c: z\mapsto z^2+ c,\quad c\in\mathbb{C}\tag{1} \] and \(d(c)\) the Hausdorff dimension of the Julia sets of (1). The authors prove that \(c\mapsto d'(c)\) tends to \(+\infty\) from the left at \({1\over 4}\) as \(({1\over 4}- c)^{d({1\over 4})-{3\over 2}}\). Hence the authors determine the speed of convergence of \(d(c)\) towards \(d({1\over 4})\) from the left. In particular, the graph of \(d\) has a vertical tangent on the left at \({1\over 4}\), which supports the numerical experiment.