Probability measures on fractal curves (probability distributions on the Vicsek fractal)

Abstract

In this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable ζ = ∑_k=0,...,∞ 23^-kε_ηk ≡ Δ_η1,...$,ηk where the indices η_k are independent random variables taking the values 0,1,2,3,4 with probabilities p_0k, p_1k, p_2k, p_3k, p_4k respectively, p_ik ≥ 0, p_ik ≥ 0, ∑_i=0,...,4p_ik = 1 for any k ∈ ℕ, ε_m = e^mπi/2 = i^m (m = 0,1,2,3) are 4th roots of unity, ε₄ = 0. We prove that the distribution of ζ is supported on a self-similar fractal curve such that the branching index of every point is equal to 2 or 4. We obtain conditions for the distribution of ζ to be of pure type: discrete or singularly continuous with respect to two-dimensional Lebesgue measure. In the case of discrete distribution, the point spectrum is described. In the case of singular distribution, we describe topological, metric and fractal properties of minimal closed support (spectrum). We study in detail the case of identically distributed indices. In particular, the mathematical expectation and variance of the random variable are found.