Singularity of the Digit Inversor for the Q3-Representation of the Fractional Part of a Real Number, Its Fractal and Integral Properties

Abstract

We introduce and study a continuous function I which is called a digit inversor for the Q₃-representation of the fractional part of a real number. This representation is determined by a probability vector.(q₀; q₁; q₂) with positive coordinates, generalizes the classical ternary representation, and coincides with this representation for q₀ = q₁ = q₂ = 1/3: The values of this function are obtained from the Q₃-representation of the argument by the following change of digits: 0 by 2; 1 by 1; and 2 by 0: The differential, integral, and fractal properties of the inversor are described. We prove that I is a singular function for q₀ ≠ q₂.