On the derivative of the Hausdorff Dimension of the Julia sets for z^2+c

TL;DR

This paper investigates how the Hausdorff dimension of Julia sets for quadratic polynomials changes as the parameter approaches a parabolic value, providing insights into the fractal geometry of Julia sets.

Contribution

It analyzes the derivative of the Hausdorff dimension function near parabolic parameters, offering new understanding of the dimension's behavior in this regime.

Findings

Characterizes the behavior of d(c) near parabolic parameters

Provides formulas for the derivative of the Hausdorff dimension

Enhances understanding of fractal geometry in complex dynamics

Abstract

Let $d(c)$ denote the Hausdorff dimension of the Julia set $J_c$ of the polynomial $f_c(z)=z^2+c$. We will investigate behavior of the function $d(c)$ when real parameter $c$ tends to a parabolic parameter.