Infinite Compositions and Complex Dynamics; Generalizing Schröder 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.

12 pages

References in corpus (3)

• Hyper-operations By Unconventional Means
• Second Order Transfer Equations; and Generalizations to Arbitrary Orders
• The Limits of a Family; of Asymptotic Solutions to The Tetration Equation