Hyers-Ulam stability of parabolic M\"obius 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} 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.