Hyers-Ulam stability of hyperbolic Möbius difference equation

Hyers-Ulam stability of the difference equation with the initial point z0 as follows zi+1 = azi+b/czi+d is investigated for complex numbers a,b,c and d where ad-bc = 1, c ? 0 and a+d ?R\[-2,2]. The stability of the sequence {zn}n?N0 holds if the initial point is in the exterior of a certain disk of which center is ?d/c . Furthermore, the region for stability can be extended to the complement of some neighborhood of the line segment between -d/c and the repelling fixed point of the map z ? az+b/cz+d. This result is the generalization of Hyers-Ulam stability of Pielou logistic equation.

Outer compositions of hyperbolic/loxodromic linear fractional transfomations

It is shown, using classical means, that the outer composition of hyperbolic or loxodromic linear fractional transformations{fn}, wherefn→f, converges toα, the attracting fixed point off, for all complex numbersz, with one possible exception,z0. I.e.,Fn(z):=fn∘fn−1∘…∘f1(z)→αWhenz0exists,Fn(z0)→β, the repelling fixed point off. Applications include the analytic theory of reverse continued fractions.