Hyers-Ulam stability of hyperbolic M\"obius difference equation

Abstract

Hyers-Ulam stability of the difference equation with the initial point $ z_0 $ as follows $$ z_{i+1} = \frac{az_i + b}{cz_i + d} $$ is investigated for complex numbers $ a,b,c $ and $ d $ where $ ad - bc = 1 $, $ c \neq 0 $ and $a + d \in \mathbb{R} \setminus [-2,2] $. The stability of the sequence $ \{z_n\}_{n \in \mathbb{N}_0} $ holds if the initial point is in the exterior of a certain disk of which center is $ -\frac{d}{c} $. Furthermore, the region for stability can be extended to the complement of some neighborhood of the line segment between $ -\frac{d}{c} $ and the repelling fixed point of the map $ z \mapsto \frac{az + b}{cz + d} $. This result is the generalization of Hyers-Ulam stability of Pielou logistic equation.