INVERSOR OF DIGITS OF TWO-BASE G–REPRESENTATION OF REAL NUMBERS AND ITS STRUCTURAL FRACTALITY

Abstract

In the paper, we introduce a new two-symbol system of representation for numbers from segment $[0;0,5]$ with alphabet (set of digits) $A=\{0;1\}$ and two bases 2 and $-2$: \[x=\dfrac{\alpha_1}{2}+\dfrac{1}{2}\sum\limits^\infty_{k=1}\dfrac{\alpha_{k+1}}{2^{k-(\alpha_1+\ldots+\alpha_k)}(-2)^{\alpha_1+\ldots+\alpha_k}}\equiv \Delta^{G}_{\alpha_1\alpha_2\ldots\alpha_k\ldots}, \;\;\; \alpha_k\in \{0;1\}.\] We compare this new system with classic binary system. The function $I(x=\Delta^G_{\alpha_1\ldots \alpha_n\ldots})=\Delta^G_{1-\alpha_1,\ldots, 1-\alpha_n\ldots}$, such that digits of its $G$--representation are inverse (opposite) to digits of $G$--representation of argument is considered in detail. This function is well-defined at points having two $G$--representations provided we use only one of them. We prove that inversor is a function of unbounded variation, continuous function at points having a unique $G$--representation, and right- or left-continuous at points with two representations. The values of all jumps of the function are calculated. We prove also that the function does not have monotonicity intervals and its graph has a self-similar structure.