Structural and self-similar properties of representations of one class of fractal functions and distributions of their values

Abstract

We consider the Q2-representation of numbers from the interval [0; 1], defined by parameters q0, q1 ∈ (0; 1), and expansion of an arbitrary number x ∈ [0; 1] by the series x = α1q1−α1 + where αk ∈ {0, 1} ≡ A, q0 + q1 = 1. We study structural, local, and global topological, metric, and fractal properties of the function defined by the equality f ϕ(x = ∆Q2 α1α2α3...αnαn+1 ) = ∆Q2 ϕ1(α1,α2)ϕ2(α2,α3)...ϕn(αn,αn+1)..., where ϕ = (ϕn) is a given sequence of functions (ϕ : (0; 1)2 → (0; 1)). For a random variable Y = F (X), where X is a random variable with a given distribution, we investigate the Lebesgue structure and spectral properties.