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

## TL;DR

This paper investigates how the Hausdorff dimension of quadratic Julia sets changes with parameters near 1/4, revealing that the dimension decreases along most directions except a specific cone, with implications for understanding fractal geometry.

## Contribution

It provides the first analysis of the directional derivatives of Julia set dimensions at the parameter 1/4, excluding the parabolic direction, and characterizes their sign behavior.

## Key findings

- Directional derivatives are negative in the closed left half-plane.
- The derivative is negative in all directions except a cone around the positive real axis.
- Computer simulations support the theoretical results.

## Abstract

Let $d(\varepsilon)$ and $\mathcal D(\delta)$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_\varepsilon(z)=z^2+1/4+\varepsilon$ and $f_\delta(z)=(1+\delta)z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\varepsilon)$ 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 $\varepsilon\in\mathbb{R}^+$ (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 $\mathbb{R}^+$.