Outer compositions of hyperbolic/loxodromic linear fractional transfomations

Abstract

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