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

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.