Properties and distributions of values of fractal functions related to $\boldsymbol {Q_2}$-representations of real numbers

A $Q_2$-representation of numbers $x\in [0;1]$ is determined by a parameter $q_0\in (0;1)$ and provides an expansion of a number $x\in [0;1]$ in the form of the following series: \[ x=\alpha _1q_{1-\alpha _1}+\sum _{k=2}^{\infty }{\left (\alpha _kq_{1-\alpha _k}\prod _{j=1}^{k-1}{q_{\alpha _j(x)}}\right )}\equiv \Delta ^{Q_2}_{\alpha _1\alpha _2\ldots \alpha _n\ldots }, \] where $\alpha _k\in \{0,1\}\equiv A$ and $q_1\equiv 1-q_0$. The structural and local as well as global topological/metric and fractal properties are studied for the function $f_{\varphi }$ defined by \begin{align*} f_{\varphi }(x)&=f_{\varphi }\left (\Delta ^{Q_2}_{\alpha _1\alpha _2\alpha _3\ldots \alpha _{n-1}\alpha _{n}\alpha _{n+1}\ldots }\right ) \\ &=\Delta ^{Q_2}_{\varphi (\alpha _1,\alpha _2)\varphi (\alpha _2,\alpha _3)\ldots \varphi (\alpha _{n-1},\alpha _{n})\varphi (\alpha _{n},\alpha _{n+1})\ldots }, \end{align*} where $\varphi$ is a given function, $\varphi \colon A^2\rightarrow A$.

For the random variable $Y=f_{\varphi }(X)$ where $X$ is a random variable with a known distribution, we study its Lebesgue structure (the discrete, absolutely continuous, and singular components) and spectral properties of the set of points of growth of the distribution function of $Y$.