One class of continuous locally complicated functions related to infinite-symbol $\Phi$-representation of numbers
M. V. Prats’ovytyĭ, Oleksandr Baranovskyi, O.I. Bondarenko, Софія Ратушняк · Matematychni Studii · 2023
In the paper, we introduce and study a massive class of continuous functions defined on the interval $(0;1)$ using a special encoding (representation) of the argument with an alphabet $ \mathbb{Z}=\{0,\pm 1, \pm 2,...\}$ and base $\tau=\frac{\sqrt{5}-1}{2}$: $\displaystyle x=b_{\alpha_1}+\sum\limits_{k=2}^{m}(b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i})\equiv\Delta^{\Phi}_{\alpha_1\alpha_2...\alpha_m(\emptyset)},\quadx=b_{\alpha_1}+\sum\limits_{k=2}^{\infty}(b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i})\equiv\Delta^{\Phi}_{\alpha_1\alpha_2...\alpha_n...},$ where $\alpha_n\in \mathbb{Z}$, $\Theta_n=\Theta_{-n}=\tau^{3+|n|}$,$b_n=\sum\limits_{i=-\infty}^{n-1}\Theta_i=\begin{cases}\tau^{2-n}, & \mbox{if } n\leq0, \\1-\tau^{n+1}, & \mbox{if } n\geq 0.\end{cases}$ The function $f$, which is the main object of the study, is defined by equalities$\displaystyle\begin{cases}f(x=\Delta^{\Phi}_{i_1...i_k...})=\sigma_{i_11}+\sum\limits_{k=2}^{\infty}\sigma_{i_kk}\prod\limits_{j=1}^{k-1}p_{i_jj}\equiv\Delta_{i_1...i_k...},\\f(x=\Delta^{\Phi}_{i_1...i_m(\emptyset)})=\sigma_{i_11}+\sum\limits_{k=2}^{m}\sigma_{i_kk}\prod\limits_{j=1}^{k-1}p_{i_jj}\equiv\Delta_{i_1...i_m(\emptyset)},\end{cases}$ where an infinite matrix $||p_{ik}||$ ($i\in \mathbb{Z}$, $k\in \mathbb N$) satisfies the conditions 1) $|p_{ik}|<1$ $\forall i\in \mathbb{Z}$, $\forall k\in \mathbb N;\quad$2) $\sum\limits_{i\in \mathbb{Z}}p_{ik}=1$ $\forall k\in\mathbb N$; 3) $0<\sum\limits_{k=2}^{\infty}\prod\limits_{j=1}^{k-1}p_{i_jj}<\infty~~\forall (i_j)\in L;\quad$4) $0<\sigma_{ik}\equiv\sum\limits_{j=-\infty}^{i-1}p_{jk}<1$ $\forall i\in \mathbb Z, \forall k\in \mathbb N.$ This class of functions contains monotonic, non-monotonic, nowhere monotonic functions and functionswithout monotonicity intervals except for constancy intervals, Cantor-type andquasi-Cantor-type functions as well as functions of bounded and unbounded variation. The criteria for the function $f$ to be monotonic and to be a function of the Cantor type as well as the criterion of nowhere monotonicity are proved. Expressions for the Lebesgue measure of the set of non-constancy of the function and for the variation of the function are found. Necessary and sufficient conditions for thefunction to be of unbounded variation are established.