Group of continuous transformations of real interval preserving tails of $G_2$-representation of numbers

In the paper, we consider a two-symbol system of encoding for real numbers with two bases having different signs ${g_0<1}$ and $g_1=g_0-1$. Transformations (bijections of the set to itself) of interval $[0,g_0]$ preserving tails of this representation of numbers are studied. We prove constructively that the set of all continuous transformations from this class with respect to composition of functions forms an infinite non-abelian group such that increasing transformations form its proper subgroup. This group is a proper subgroup of the group of transformations preserving frequencies of digits of representations of numbers.