Permutable power series and regular iteration

Abstract

If f(z) is a power series convergent for |z| < ρ, where ρ > 0, then f(z) is said to have a fixpoint of multiplir a1 at z = 0. In the (local) iteration of f(z) one studies the sequence {fn(z)}, n = 0, 1, 2, … in a neighbourhood of z = 0, fn(z) benig defined by For many values of the multiplier a1, including 0 < |a1| < 1 and |a1| > 1, the local iteration of f(z) is completely mastered by the introduction of Schröder's functional equation.