On the path properties of a lacunary power series

A power series f(z) which converges in D = {|z| < 1} maps the radii [0, ζ) onto paths Γ(ζ), ζ ∈ T = ∂D. These are studied under several aspects in the case of the special lacunary series f(z) = z +z²+z⁴+z⁸+.... First, the Γ(ζ) are considered as random functions on the probability space (T,B, mes/2π), where B is the σ-algebra of Borel sets and mes the Lebesgue measure. Then analytical properties of the Γ(ζ) are discussed which hold on subsets A of T with Hausdorff dimension 1 in spite of mesA = 0. Furthermore, estimates of the derivative of f and of the arc length of sections of the Γ(ζ) are given. Finally, these results are used to derive connections between the distribution of critical points of f and the overall behaviour of the paths.