On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at \(-3/4\)

Let \(d(\delta )\) $d(\delta )$ denote the Hausdorff dimension of the Julia set of the polynomial \(f_\delta (z)=z^2-3/4+\delta \) $f_{\delta }(z)=z^2-3/4+\delta$ . In this paper, we deal with the function \(\delta \mapsto d(\delta )\) $\delta \mapsto d(\delta )$ for parameters \(\delta \) $\delta$ close to 0. Obviously, \(\delta =0\) $\delta =0$ corresponds to \(c=-3/4\) $c=-3/4$ in the case of the classical family \(p_c(z)=z^2+c\) $p_c(z)=z^2+c$ , and the polynomial \(f_0(z)\) $f_0(z)$ has a parabolic fixed point with two petals. The function \(d(\delta )\) $d(\delta )$ is not continuous at 0. However, it follows from the result proved by C. McMullen that \(d(\delta )\) $d(\delta )$ is continuous inside cones, avoiding the boundary of the Mandelbrot set \(\mathcal M\) $\mathcal{M}$ (except the vertex at 0). So, in particular, \(d(\delta )\) $d(\delta )$ is continuous along any direction inside \(\mathcal M\) $\mathcal{M}$ that converges to 0. We carefully study the behavior of \(d(\delta )\) $d(\delta )$ along these directions. That is, we consider all directions landing at 0 except two imaginary ones, which are related to the parabolic implosion phenomenon. Under numerically verified assumption \(d(0)<4/3\) $d(0)<4/3$ (i.e., \(\frac{3}{2}d(0)-2<0\) $\frac{3}{2}d(0)-2<0$ ), we prove that for any direction, the directional derivative along this direction, divided by \(|\delta |^{\frac{3}{2}d(0)-2}\) ${|\delta |}^{\frac{3}{2}d(0)-2}$ , tends to a constant. We give a formula for this constant (depending on the direction) with accuracy to a universal multiplicative positive factor. Therefore, this formula allows us to verify numerically whether, in a given direction, the function \(d(\delta )\) $d(\delta )$ is increasing or decreasing.