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

Hyers-Ulam of the sequence $\{z_n{\}}_{n\in \mathbb{N}}$ satisfying the difference equation $z_{i+1}=g(z_i)$ where $g(z)=\frac{az+b}{cz+d}$ with complex numbers $a$, $b$, $c$ and $d$ is defined. Let $g$ be loxodromic M"obius map, that is, $g$ satisfies that $ad-bc=1$ and $a + d \in $\mathbb{C}$ \setminus [-2,2] $. Hyers-Ulam stability holds if the initial point of $\{z_n{\}}_{n\in \mathbb{N}}$ is in the exterior of avoided region, which is the union of the certain disks of $g^{-n}(\mathrm{\infty })$ for all $ n \in $\mathbb{N}$ $.