Singular monotone functions, which are determined by convergent series and double stochastic matrix

Abstract

The paper considers functions, which are defined by $$F(x)=\overline{\Delta}_{a_1(x)a_2(x)\ldots a_n(x)\ldots}^2={\Delta}_{a_1(x)a_2(x)\ldots a_n(x)\ldots}=y,$$
where $\overline{\Delta}_{a_1a_2\ldots a_n\ldots}^2=\frac{2}{3}+\frac{\alpha_1}{(-2)^1}+\frac{\alpha_2}{(-2)^2}+\frac{\alpha_3}{(-2)^3}+\ldots$ is the binary nega-positional representation of the number on the interval $[0; 1]$,
${\Delta}_{a_1a_2\ldots a_n\ldots}$ is the Markov representation determined by the positive doubly-stochastic matrix
$$\|p_{ik}\|=\begin{pmatrix}
p_{00} & p_{01}\\
p_{10} & p_{11}
\end{pmatrix}.$$
The singularity and self-similar properties of the functions are established. Functional relations between them are found.