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

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Properties of the family of continuous functions preserving a $Q_s$-digit with accumulation
Natalya Vasylenko, Irina Zamriy

Last modified: 2018-09-22


Let $Q_s=\{q_0,q_1,\ldots,q_{s-1}\}$ be a fixed set of positive real number such that $q_0+q_1+\ldots+q_{s-1}=1$. It is know that for any number $x\in [0;1]$ there exists sequence $(\alpha_n)$, $\alpha_n\in A_s=\{0,1,\ldots, s-1\}$, such that\[x=\beta_{\alpha_1(x)}+\sum\limits_{k=2}^{\infty}\left(\beta_{\alpha_k(x)}\prod\limits_{j=1}^{k-1} q_{\alpha_j(x)}\right)\equiv\Delta^{Q_s}_{\alpha_1(x) \alpha_2(x) \ldots \alpha_k(x) \ldots},\]

where $\alpha_k(x)\in A_s$,  $\beta_0=0$, $\beta_i=\sum\limits_{j=0}^{i-1}q_j$.

We consider a family of functions $f$ satisfying conditions:
\[ y=f(x)=f(\Delta^{Q_s}_{\alpha_1(x) \alpha_2(x) \ldots \alpha_k(x) \ldots})=\Delta^{Q_s}_{\delta_1 \delta_2 \ldots \delta_k \ldots}, \:\: \text{where}\] \[  \delta_k =\left\{    \begin{aligned}    & \varphi_k(\alpha_1(x), \alpha_2(x), \ldots, \alpha_{k}(x))\:\:\text{if }  \: \alpha_k(x)\in A_s\setminus\{m\},\\    & m\:\:\text{if } \alpha_k(x)\in A_s;    \end{aligned} \right. \]and the sequence of functions $\varphi_k$ is given.
In the talk we study structural, differential and fractal properies of function $f$.