SINGULARITY OF THE DIGIT INVERSOR FOR THE Q_3-REPRESENTATION OF THE FRACTIONAL PART OF A REAL NUMBER, ITS FRACTAL AND INTEGRAL PROPERTIES

摘要

We introduce and study a continuous function Ⅰ which is called a digit inversor for the Q_3-representation of the fractional part of a real number. This representation is determined by a probability vector (q_0, q_1, q_2) with positive coordinates, generalizes the classical ternary representation, and coincides with this representation for q_0 = q_1 = q_2 = 1/3. The values of this function are obtained from the Q_3-re-presentation 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_0 ≠ q_2.