Groups of continuous transformations of the unit interval and the real line

This paper is devoted to the study of $g$-representations of real numbers in the unit interval that are topologically equivalent to the classical $s$-adic representation, and continuous transformations of the unit interval associated with such number representations. The transformations under consideration are obtained by gluing together the left and right shift operators of the $g$-representation. While these transformations are piecewise linear in the case of the $s$-adic representation, this property does not hold in general for arbitrary $g$-representations. It is constructively proved that the set of all continuous transformations of the unit interval preserving the digit frequencies of the $g$-representation, equipped with the operation of composition, forms a noncommutative group of cardinality continuum. A continuum subgroup of this group is formed by the transformations preserving the tails of the $g$-representations of numbers.