Inversor of digits $Q^∗_2$-representative of numbers

Abstract

We consider structural, integral, differential properties of function defined by equality$$I(\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=\Delta^{Q_2^*}_{[1-\alpha_1][1-\alpha_2]...[1-\alpha_n]...}, \quad \alpha_n\in A\equiv\{0,1\}$$for two-symbol polybasic non-self-similar representation of numbers of closed interval $[0;1]$ that is a generalization of classic binary representation and self-similar two-base $Q_2$-representation.For additional conditions on the sequence of bases, singularity of the function and self-affinity of the graph are proved.Namely, the derivative is equal to zero almost everywhere in the sense of Lebesgue measure.The integral of the function is calculated.