Transformations of \((0,1]\) preserving tails of \(\Delta^{\mu}\)-representation of numbers

Abstract

Let \(\mu\in (0,1)\) be a given parameter, \(\nu\equiv 1-\mu\). We consider \(\Delta^{\mu}\)-representation of numbers \(x=\Delta^{\mu}_{a_1a_2\ldots a_n\ldots}\) belonging to \((0,1]\) based on their expansion in alternating series or finite sum in the form:
\[
x=\sum_n(B_{n}-{B'_n})\equiv \Delta^{\mu}_{a_1a_2\ldots a_n\ldots},
\]
where
\(B_n=\nu^{a_1+a_3+\ldots+a_{2n-1}-1}{\mu}^{a_2+a_4+\ldots+a_{2n-2}},\)

\({B^{\prime}_n}=\nu^{a_1+a_3+\ldots+a_{2n-1}-1}{\mu}^{a_2+a_4+\ldots+a_{2n}},\) \(a_i\!\in\! \mathbb{N}\).

This representation has an infinite alphabet \(\{1,2,\ldots\}\), zero redundancy and \(N\)-self-similar geometry.

In the paper, classes of continuous strictly increasing functions preserving ``tails'' of \(\Delta^{\mu}\)-representation of numbers are constructed. Using these functions we construct also continuous transformations of \((0,1]\). We prove that the set of all such transformations is infinite and forms non-commutative group together with an composition operation.