Hyers-Ulam stability of hyperbolic Möbius difference equation

Abstract

Hyers-Ulam stability of the difference equation with the initial point $z_0$ as follows $$z_{i+1}=\frac{a{z}_i+b}{c{z}_i+d}$$ is investigated for complex numbers $a,b,c$ and $d$ where $ad-bc=1$, $c\ne 0$ 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.

30 pages, 6 figures