Hyers-Ulam stability of parabolic Möbius difference equation

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} Möbius map. Let $\{b_n{\}}_{n\in {\mathbb{N}}_0}$ be the solution of the parabolic Möbius 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.

4 figures, 23 pages