A group of continuous transformations of a region $[0; 1]$ that preserve the frequency of the digits in $Q_s$-representation of a number

Abstract

In the paper, we propose to solve the problem of construction a set of a continuous transformation of the segment $[0;1]$, which keeps the figure frequency of a $????_????$ real number representation for any natural $???? >1$. The group properties of that set are studied in the article. Sets of functions which keeps figure frequency and functions which keepping tails of the $????_????$ number representation are presented in the paper.