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

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

• d \in $\mathbb{R}$ \setminus [-2,2] $\mathrm{.}Thestabilityofthesequence$ {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.