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

Abstract

Let d(ɛ) and denote the Hausdorff dimension of the Julia sets of the polynomials pɛ(z) = z2 + 1/4 + ɛ and fδ(z) = (1 + δ)z + z2 respectively. In this paper we will study the directional derivative of the functions d(ɛ) and along directions landing at the parameter 0, which corresponds to 1/4 in the case of family z2 + c. We will consider all directions, except the one (or two imaginary directions in the δ 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°) around .