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

摘要

We study continuous, strictly monotonic functions preserving tails of an A-continued fraction representation of real numbers.We construct continuous transformations of [ 1 2 ; 1 ] {[rac{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).