On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at -2

Abstract

Let $d(\delta)$ denote the Hausdorff dimension of the Julia set of the polynomial $f_\delta(z)=z^2-2+\delta$. In this paper we will study the directional derivative of the function $d$ along directions landing at the parameter $0$, which corresponds to $-2$ in the case of family $p_c(z)=z^2+c$. We will consider all directions, except the one $\delta\in\mathbb{R}^+$, which is inside the Mandelbrot set. We will prove asymptotic formula for the directional derivative of $d$. Moreover, we will see that the derivative is negative for all directions in the closed left half-plane. Computer calculations show that it is negative except a cone (with opening angle approximately $74^\circ$) around $\mathbb{R}^+$.