Про DP-властивостi фрактальних ймовiрнiсних мiр з незалежними Q-символами

Abstract

We study continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP- transformations”) of every subset of R1. It is shown that the problem of investigation of continuous DP-transformations of the real line is equivalent to the problem of studying the DP-properties of strictly increasing probability distribution functions on a unit interval. Apply- ing the multilevel fractal analysis of singularly continuous probability measures with independent Q-digits, we found sharp (necessary and sufficient) conditions for the Hausdorff-Besicovitch dimension preservation under the corresponding distribution functions.