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

TL;DR

This paper investigates how the Hausdorff dimension of quadratic Julia sets changes with parameters near -2, providing asymptotic formulas for directional derivatives and revealing their negativity in most directions outside the positive real axis.

Contribution

It introduces the first analysis of the directional derivative of the Hausdorff dimension function at the parameter -2 for quadratic Julia sets, including asymptotic formulas and numerical insights.

Findings

Directional derivatives are negative in all directions in the closed left half-plane.

Asymptotic formulas for the derivatives are established.

Numerical evidence suggests the derivative is negative except within a 74° cone around the positive real axis.

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 $\mathbb{R}^+$.