Transformations and functions that preserve the asymptotic mean of digits in the ternary representation of a number

In the paper we study transformations of the interval $[0;1)$ and functions that preserve the asymptotic mean $r$ of the digits in the $s$–adic representation of a number $x$, $$r(x)=\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\sum _{i=1}^n{\alpha }_i(x)$$ specify the necessary and sufficient conditions for a transformation to belong to this class