Аннотация:
The paper is devoted to the study of connections between fractal
properties of one-dimensional singularly continuous probability measures and the preservation of the Hausdorff dimension of any subset
of the unit interval under the corresponding distribution function.
Conditions for the distribution function of a random variable with
independent $Q$-digits to be a transformation preserving the Hausdorff dimension (DP-transformation) are studied in details. It is
shown that for a large class of probability measures the distribution function is a DP-transformation if and only if the corresponding
probability measure is of full Hausdorff dimension.
Ключевые слова:Singularly continuous probability distributions, Hausdorff dimension of probability measures, Hausdorff-Billingsley dimension, fractals, DP-transformations.