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 a 1 at z = 0. In the (local) iteration of f ( z ) one studies the sequence { f n (z) }, n = 0, 1, 2, … in a neighbourhood of z = 0, f n (z) benig defined by For many values of the multiplier a 1 , including 0 < | a 1 | < 1 and | a 1 | > 1, the local iteration of f(z) is completely mastered by the introduction of Schröder's functional equation .