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We consider $Q_s$-representation of numbers$x\in[0,1]$ defined by parameters $q_0,q_1,...,q_{s-1}\in(0;1)$ and expansion of numbers $x\in[0,1]$ in series\[x=\alpha_1q_{1-\alpha_1}+\sum\limits_{k=2}^{\infty}{(\alpha_kq_{1-\alpha_k}\prod\limits_{j=1}^{k-1}{q_{\alpha_j(x)}})}\equiv\Delta^{Q_s}_{\alpha_1\alpha_2\ldots\alpha_n\ldots},\]where $\alpha_k\in\{0,1,...s-1\}\equiv A$, $q_0+q_1+...+q_{s-1}=1$. We study structural, local and global topological, metric andfractal properties of the function defined by equality\[f_{\varphi}(x)=f_{\varphi}(\Delta^{Q_s}_{\alpha_1\alpha_2\alpha_3...\alpha_{n-1}\alpha_{n}\alpha_{n+1}...})=\Delta^{Q_s}_{\varphi(\alpha_1,\alpha_2)\varphi(\alpha_2,\alpha_3)...\varphi(\alpha_{n-1},\alpha_{n})\varphi(\alpha_{n},\alpha_{n+1})...},\]where $\varphi$ is a given function $(\varphi: A^2\rightarrow A)$.
For random variable $Y=F(X)$, where $X$ is a random variable with a given distribution, Lebesgue structure andspectral properties are studied.