У статті розглядається представлення дійсних чисел рядами Кантора та випадкова величина з незалежними символами вищезгаданого представлення. Функція розподілу цієї величини є неперервною та строго зростаючою на відрізку [0; 1], тому вона задає перетворення цього відрізка. У статті наводиться критерій (для широкого класу рядів Кантора) того, щоб це перетворення зберігало пакувальну фрактальну розмірність.
The Hausdorff dimension dimH is the most famous fractal dimension. It is well known that the determination of this dimension is a rather non-trivial problem for many sets and measures.
The packing dimension dimP can be considered as an alternative fractal dimension [31, 10]. It has been introduced only in 1980-s but it is widely known and useful in the study of fractal sets and measures.
The dimension preserving transformations is the useful approach for Hausdorff-Besicovitch dimension researching [2].
Definition. A transformation f of space M is called dimension preserving if
The aim of this paper is to develop the similar approach for the packing dimension researching. This approach is based on the packing dimension preserving transformations notion.
J. Li [18] has proven some sufficient conditions for distribution functions of random variables with independent Q-digits to be in PDP -class. More precisely, J.Li has proven the next theorem:
Theorem. Let be the distribution function of the random variable with independent
Q-representation. In remark 4.2 at the end of article [18] one can read: The conditions inf qij = q* > 0 and infj j pij = p* > 0 play an important role in the proof of the theorem. Open question: What can we say about the topic if we remove these conditions?.
S. Albeverio, M. Pratsiovytyi and G. Torbin removed condition inf pij = p* > 0 in similar situation for DP -transformations in [3].
The main result of the paper is the sharp condition of the F to be PDP if F is the distribution function for random variable with independent Cantor series digits and condition is removed.
