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

TL;DR

This paper investigates the Hyers-Ulam stability of solutions to parabolic Möbius difference equations, concluding that such sequences lack this stability property.

Contribution

It establishes that solutions to parabolic Möbius difference equations do not exhibit Hyers-Ulam stability, filling a gap in the understanding of stability for these complex dynamical systems.

Findings

Sequences have no Hyers-Ulam stability

Stability properties differ from other Möbius maps

Results contribute to complex dynamics theory

Abstract

The linear fractional map $ g(z) = \frac{az+ b}{cz + d} $ on the Riemann sphere with complex coefficients $ ad-bc = 1 $ is and $ a+d = \pm 2 $, then $ g $ is called {\em parabolic} M\"obius map. Let $ \{ b_n \}_{n \in \mathbb{N}_0} $ be the solution of the parabolic M\"obius difference equation $ b_{n+1} = g(b_n) $ for every $ n \in \mathbb{N}_0 $. We show that the sequence $ \{ b_n \}_{n \in \mathbb{N}_0} $ has no Hyers-Ulam stability.