One continuum class of fractal functions defined in terms of $Q_s^{\ast }$-representation

Abstract

In the paper we study a class $F$ of multiparameter functions defined in terms of a polybasic $s$-adic $Q_s^{\ast }$-representation of numbers by \begin{equation*} f_a\bigl(x=Δ^{Q^{*}_s}_{α_1α_2\ldotsα_n\ldots}\bigr) = Δ^{Q^{*}s}_{|a_1-α_1|\,|a_2-α_2|\,\ldots\,|a_n-α_n|\ldots}, \end{equation*} where $(a_n)$ is the sequence of digits for $s$-adic representation of the parameter $a\in [0,1]$, and \begin{equation*} Δ^{Q^{*}_s}_{α_1α_2\ldotsα_n\ldots}= β_{α_1 1}+ \sum_{n=2}^{\infty} \left( β_{α_n n} \prod_{j=1}^{n-1} q_{α_j j} \right) \end{equation*} is the $Q_s^{\ast }$-representation of real numbers generated by a positive stochastic matrix $\Vert q_{ij}\Vert$ with ${\hat{I}}_{\hat{I}{\pm }_{n}n}^2=\sum _{i=0}^{\hat{I}{\pm }_n-1}{q}_{in}$. In this paper we investigate the continuity of the function $f_a$ on the sets of $Q_s^{\ast }$-binary and $Q_s^{\ast }$-unary numbers. We prove that the functions in this class are continuous on the set of numbers with a unique $Q_s^{\ast }$-representation. Furthermore, we show that except for $f_0$ and $f_1$, all functions have a countable set of discontinuities at $Q_s^{\ast }$-binary points. We classify the topological types of the value sets of $f_a$ depending on the parameter $a$. We prove that, if the value set is of Cantor type, then it is zero-dimensional. We describe the structural properties of the level sets of $f_a$ in terms of the digits of the $s$-adic representation of $a$. In particular, we establish that a level set of the function $f_a$ can be an empty set, a finite set, or a continuum. For certain values of $s$ we provide examples of fractal level sets and calculate its fractal dimensions.