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

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.