Uncountable group of continuous transformations of unit segment preserving tails of Q_2-representation of numbers

Abstract

We consider two-base Q2-representation of numbers of segment [0; 1]: (Formula Presented), which is defined by two bases q0 ∊ (0; 1), q1 = 1 - q0 and an alphabet A = {0, 1}, (αn) ∊ A × A × . . . . It is a generalization of classic binary representation (q0 = 1/2 ). In the article we prove that the set of all continuous bijections of segment [0; 1] preserving “tails” of Q2-representation of numbers forms an uncountable non-abelian group with respect to composition such that it is a subgroup of the group of continuous transformations preserving frequencies of digits of Q2-representation of numbers. Construction of such transformations (bijections) is based on the left and right shift operators for digits of Q2-representation of numbers.