Hyers-Ulam stability of loxodromic Möbius difference equation

Abstract

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 Möbius 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 $n\in \mathbb{N}$.

34 pages, 5 figures. arXiv admin note: text overlap with arXiv:1708.08662