# Hyers-Ulam stability of elliptic M\"obius difference equation

## TL;DR

This paper investigates the stability of solutions to elliptic M"obius difference equations, concluding that such sequences lack Hyers-Ulam stability under specified conditions.

## Contribution

It establishes that solutions to elliptic M"obius difference equations do not exhibit Hyers-Ulam stability, clarifying the stability behavior of these complex dynamical systems.

## Key findings

- Sequences have no Hyers-Ulam stability.
- Stability depends on elliptic M"obius map properties.
- Results contribute to complex dynamics and stability theory.

## Abstract

The linear fractional map $ f(z) = \frac{az+ b}{cz + d} $ on the Riemann sphere with complex coefficients $ ad-bc \neq 0 $ is called M\"obius map. If $ f $ satisfies $ ad-bc=1 $ and $ -2<a+d<2 $, then $ f $ is called $\textit{elliptic}$ M\"obius map. Let $ \{ b_n \}_{n \in \mathbb{N}_0} $ be the solution of the elliptic M\"obius difference equation $ b_{n+1} = f(b_n) $ for every $ n \in \mathbb{N}_0 $. Then the sequence $ \{ b_n \}_{n \in \mathbb{N}_0} $ has no Hyers-Ulam stability.