Сім'я функцій, які зберігають цифру $Q_s$-зображення чисел

Abstract

We consider a polybase $Q_s$-representation $\Delta^{Q_s}_{\alpha_1(x) \alpha_2(x) \ldots \alpha_k(x)\ldots}$ of number $x$ that is a generalizing of classic $s$-adic representation: $x=\sum\limits_{k=1}^{\infty} s^{-k}\alpha_k(x)=\Delta^{s}_{\alpha_1(x)\alpha_2(x) \ldots \alpha_k(x) \ldots}$, where
$\alpha_k(x)\in A_s\equiv\{0,1,\ldots,s-1\}$.
In the paper, we study continuum class of functions defined on $[0;1]$ and preserving one of the digits of $Q_s$-representation, namely:
\[
f(\Delta^{Q_s}_{\alpha_1 \alpha_2 \ldots \alpha_k \ldots})=\Delta^{Q_s}_{\delta_1 \delta_2 \ldots \delta_k \ldots}, \:\: \text{where } \alpha_k, \delta_k\in A_s,
\]
\[
\Delta^{Q_s}_{\alpha_1 \alpha_2 \ldots \alpha_k\ldots}=\beta_{\alpha_1(x)}+\sum\limits_{k=2}^{\infty} \Bigl(\beta_{\alpha_k(x)}\prod\limits_{j=1}^k q_{\alpha_j(x)}\Bigr)
\]
and
$\delta_k = \varphi_k(\alpha_1(x), \alpha_2(x), \ldots, \alpha_k(x))$ but $\delta_k=m$ if and only if $\alpha_k(x)=m$, where $m$ is a fixed digit of alphabet $A_s$.

For some representatives of various subclasses of the family of functions preserving digit $m$ of alphabet $A_s$, self-similar and fractal properties are studied in details.