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

TL;DR

This paper uses infinite compositions to solve generalized functional equations related to complex dynamics, extending classical Schr"oder and Abel functions, and explores their applications to dynamical orbit properties.

Contribution

It introduces a novel approach to solving complex functional equations via infinite compositions, generalizing classical methods and analyzing their dynamical implications.

Findings

Solved general equations using infinite compositions

Described conditions for the equations' applicability

Connected solutions to dynamical orbit behavior

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.