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

Abstract

Let d(δ) denote the Hausdorff dimension of the Julia set of the polynomial fδ(z)=z2−2+δ.In this paper, we will study the directional derivative of the function δ↦d(δ) along directions landing at the parameter 0, which corresponds to −2 in the case of family pc(z)=z2+c. We will consider all directions except the one δ∈R+, which is inside the Mandelbrot set.We will prove an asymptotic formula for the directional derivative of d(δ). 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 for a cone (with an opening angle of approximately 74∘) around R+.