Fractal probability distributions and transformations preserving the Hausdorff-Besicovitch dimension

This article is devoted to the development of a general theory of transformations of R-n, called dimension-preserving (DP) transformations, which preserve the Hausdorff-Besicovitch dimension of arbitrary subsets. The main attention is given to continuous transformations of R and [0, 1]. A class of distribution functions of random variables with independent s-adic digits is studied in detail. It is proved that any absolutely continuous function from the previously mentioned class is a DP function, despite the fact that it may have a very complicated local structure. Necessary, respectively, sufficient conditions for dimension preservation are also given for singular functions. Relations between the entropy of transformations and their DP properties are investigated. Examples and counterexamples are provided,and some applications are discussed.