Singularity of the digit inversor for the [Q.sub.3]--representation of the fractional part of a real number, its fractal and integral properties

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2016-06, Vol.215 (3), p.323
• Main Authors: Zamrii, I.V, Prats'ovytyi, M.V
• Format: Article
• Language: English
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-016-2841-y

We introduce and study a continuous function I which is called a digit inversor for the [Q.sub.3]-representation of the fractional part of a real number. This representation is determined by a probability vector ([q.sub.0], [q.sub.1], [q.sub.2]) with positive coordinates, generalizes the classical ternary representation, and coincides with this representation for [q.sub.0] = [q.sub.1] = [q.sub.2] = 1/3. The values of this function are obtained from the [Q.sub.3]-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 [q.sub.0] [not equal to] [q.sub.2].