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] which is defined by two bases q0 ∈ (0;1), q1 = 1-q0 and 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.

Keywords

• two-symbol system of encoding (representation) of numbers
• $q_2$-representation of numbers
• tail set
• bijection preserving tails of representation of numbers
• group of continuous transformations of a unit segment