Continuous distributions whose functions preserve tails of an 𝐴-continued fraction representation of numbers

Abstract

Abstract We study continuous, strictly monotonic functions preserving tails of an A-continued fraction representation of real numbers. We construct continuous transformations of {[\frac{1}{2};1]} using these functions. They determine the probability distributions on this interval. It is proved that the set of all transformations is infinite and forms a non-commutative group with respect to the operation of a composition (superposition). Thus, a group of probability distributions is constructed on the interval. These distributions have non-trivial local properties (self-similar structure).