Infinite Compositions and Complex Dynamics; Generalizing Schr\"{o}der and Abel Functions

Abstract

Using infinite compositions, we solve the general equations $P(\lambda w) = p(w)f(P(w))$ for holomorphic functions $p$ and $f$. We describe the situations in which this equation is palpable; and their effectiveness at describing dynamical properties of the orbit $f^{\circ n}(z)$. We similarly make a change of variables to study a generalized form of the Abel equation, $F(s+1) = u(s)f(F(s))$. This paper is intended as a more in depth examination of work done previously in our last paper--The Limits of a Family; Of Asymptotic Solutions to the Tetration Equation.