Probability distributions with independent $Q$-symbols and transformations preserving the Hausdorff dimension

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.