Institute of Mathematics Conferences, Sixth International Conference on Analytic Number Theory and Spatial Tessellations

Font Size: 
Fractal properties of one class of nonmonotonic singular functions of Cantor's type
Svitlana Vaskevych

Last modified: 2018-09-22


Let $A_3=\{0,1,2\}$ be an alphabet of ternary numeral system, $L_3=A_3\times A_3\times ...$ be a space of sequences of the alphabet, $Q_3^*=\|q_{ik}\|$ be an infinite positive stochastic matrix ($i=0,1,2$; $k=1,2,...$), namely

$$Q^*_3=\left(              \begin{array}{ccccc}                q_{01} & q_{02} & \ldots & q_{0k} & \ldots \\                q_{11} & q_{12} & \ldots & q_{1k} & \ldots \\                q_{21} & q_{22} & \ldots & q_{2k} & \ldots \\              \end{array}      \right),$$

where $q_{ik}>0$, $q_{0k}+q_{1k}+q_{2k}=1$;  the matrix is define $Q^*_3$-representation of numbers:$ x=\beta_{\alpha_1(x)1}+\sum\limits^{\infty}_{k=2}\left[\beta_{\alpha_k(x)k}\prod\limits^{k-1}_{j=1}q_{\alpha_j(x)j}\right]\equiv\Delta^{Q^*_3}_{\alpha_1\alpha_2...\alpha_k...},$where $\beta_{0k}=0$, $\beta_{1k}=q_{0k}=\beta_{0k}+q_{0k}$, $\beta_{2k}=q_{0k}+q_{1k}=\beta_{1k}+q_{1k}$.

Note that if $q_i\equiv q_{i1}=q_{i2}=...=q_{ik}=...$, where $i\in A_3$, $k\in N$, we have $Q_3$-representation determined by  positive numbers $q_0$, $q_1$, $q_2$, and if $q_0=q_1=q_2=\dfrac{1}{3}$ we have classic ternary representation.

Let also $(\varepsilon_k)$ be a sequence of real numbers such that $0\leqslant\varepsilon_k\leqslant 1$; let $g_{0k}=\dfrac{1+\varepsilon_k}{3}=g_{2k}$, $g_{1k}=\dfrac{1-2\varepsilon_k}{3}$, $k\in N$.

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality

$$f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j(x)j}\right] \equiv \Delta^{G^*_3}_{\alpha_1\alpha_2\ldots\alpha_k\ldots},$$

where $\delta_{0k}=0$, $\delta_{1k}=g_{0k}=\delta_{0k}+g_{0k}$, $\delta_{2k}=g_{0k}+g_{1k}=\delta_{1k}+g_{1k}$.

In the talk we pay attention to the criteria of strict monotonicity, non monotonicity and nowhere monotonicity, non-differentiability and singularity of the functions. We study properties of the level sets of the functions and its graph.