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}, where fn ￠ ’f, converges to ±, 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) ￠ ’ ±When z0 exists, Fn(z0) ￠ ’ 2, the repelling fixed point of f. Applications include the analytic theory of reverse continued fractions.