\(R^*\)-transformers and fractal curves
\(R\)-transformers process binary representation of real numbers and, as it has been already shown by \textit{L. P. Lisovik} [Kibern. Sist. Anal. 3, No. 3, 11-22 (1994; Zbl 0868.28004)], make it possible to specify a continuous nowhere-differentiable function, a Peano curve, a singular strictly increasing function, and some fractal curves. In the current paper the authors define pseudo-synchronous \(R\)-transformers and \(R^*\)-transformers. It is shown that the class of functions defined by finite pseudo-synchronous \(R\)-transformers is closed with respect to the operations of addition, multiplication by a rational number, and modulus taking. The authors argue that \(R^*\)-transformers (allowing both digits themselves and the radix of the numeration system to be fractional numbers in representations of real numbers appearing on the output tape) are more convenient in representing continuous real functions and fractal curves than the \(R\)-transformers. The authors do not compare their approach with conventional approaches widely used to define fractal sets and based on iterated function systems.
scientific article; zbMATH DE number 1439546