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 $g^{-n}(\infty)$ for all $n \in \mathbb{N}$.