Some Sets of Relative Stability Under Perturbations of Branched Continued Fractions with Complex Elements and a Variable Number of Branches

Abstract

The present paper deals with the investigation of the conditions under which infinite branched continued fractions are stable under perturbations of their elements. We establish the formulas for the relative errors of the approximants of branched continued fractions with complex partial denominators and numerators that are equal to one. By using the technique of the sets of elements and the corresponding sets of values of the tails of approximants, we construct the sets of relative stability under perturbations, namely, the angular sets and the sets representing the exterior domains of circles on the even floors of the fraction and half planes on its odd floors. We also establish estimates for the relative errors of approximants of these branched continued fractions.