One continuum class of fractal functions defined in terms of $Q^*_s$-representation

TL;DR

This paper introduces a class of fractal functions based on $Q^*_s$-representations, analyzing their continuity, topological properties, and fractal dimensions of level sets, with implications for understanding their structural complexity.

Contribution

It provides a novel analysis of a continuum class of fractal functions defined via $Q^*_s$-representations, including their continuity, topological classification, and fractal properties.

Findings

Functions are continuous on numbers with unique $Q^*_s$-representation.

All but two functions have countably many discontinuities at $Q^*_s$-binary points.

Level sets can be empty, finite, or fractal, with some examples of fractal dimensions provided.

Abstract

In the paper we study a class $F$ of multiparameter functions defined in terms of a polybasic $s$-adic $Q^{*}_{s}$-representation of numbers by \begin{equation*} f_a\bigl(x=\Delta^{Q^{*}_s}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}\bigr) = \Delta^{Q^{*}s}_{|a_1-\alpha_1|\,|a_2-\alpha_2|\,\ldots\,|a_n-\alpha_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*} \Delta^{Q^{*}_s}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}= \beta_{\alpha_1 1}+ \sum_{n=2}^{\infty} \left( \beta_{\alpha_n n} \prod_{j=1}^{n-1} q_{\alpha_j j} \right) \end{equation*} is the $Q^{*}_{s}$-representation of real numbers generated by a positive stochastic matrix $\|q_{ij}\|$ with $\beta_{\alpha_n n}=\sum\limits_{i=0}^{\alpha_n-1} q_{in}$. In this paper we investigate the continuity of the function $f_a$ on the sets of $Q^{*}_{s}$-binary and $Q^{*}_{s}$-unary numbers. We prove that the functions in this class are continuous on the set of numbers with a unique $Q^{*}_{s}$-representation. Furthermore, we show that except for $f_0$ and $f_1$, all functions have a countable set of discontinuities at $Q^{*}_{s}$-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.