On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4

Abstract

Let $d(\epsilon )$ and $\mathcal{D}(\delta )$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_{\epsilon }(z)=z^2+1/4+\epsilon$ and $f_{\delta }(z)=(1+\delta )z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\epsilon )$ and $\mathcal{D}(\delta )$ along directions landing at the parameter $0$, which corresponds to $1/4$ in the case of family $z^2+c$. We will consider all directions, except the one $\epsilon \in {\mathbb{R}}^{+}$ (or two imaginary directions in the $\delta$ parametrization) which is outside the Mandelbrot set and is related to the parabolic implosion phenomenon. We prove that for directions in the closed left half-plane the derivative of $d$ is negative. Computer calculations show that it is negative except a cone (with opening angle approximately $150^{\circ }$) around ${\mathbb{R}}^{+}$.