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

Tetiana M. Isaieva ; Mykola V. Pratsiovytyi

Algebra and discrete mathematics, Tome 22 (2016) no. 1, pp. 102-115

Résumé

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.

Détail

Keywords: $\Delta^{\mu}$-representation, cylinder, tail set, function preserving “tails” of $\Delta^{\mu}$-representation of numbers, continuous transformation of $(0,1]$ preserving “tails” of $\Delta^{\mu}$-representation of numbers
Mots-clés : group of transformations.