Asymptotic Solutions of the Tetration Equation

TL;DR

This paper constructs a family of holomorphic functions that approximate iterated exponentials asymptotically, providing insights into the dynamics of exponential maps and fractional iteration.

Contribution

It introduces a new class of functions with asymptotic exponential behavior, linking them to exponential dynamics and Abel equations.

Findings

Functions approximate iterated exponentials in right half-plane

Provides series expansions converging in half-plane

Connects to holomorphic Abel functions and exponential dynamics

Abstract

In this report we construct a family of holomorphic functions $\beta_{\lambda,\mu} (s)$ which behave asymptotically like iterated exponentials as $|s| \to \infty$ in the right half plane. Each $\beta_{\lambda,\mu}$ satisfies a convenient functional relationship with nested exponentials; and has a series expansion that converges in a half-plane. They provide a nearness to the dynamics of the map $e^{\mu z} : \mathbb{C}\to\mathbb{C}$ and behave asymptotically as a fractional iteration would behave. These objects are used to describe the various orbits of the exponential function. We describe where Abel equations are feasibly constructed from $\beta$. Where there exists wildly holomorphic functions with period $2 \pi i / \lambda$ that are holomorphic Abel functions of the form $t(s+1) = e^{\mu t(s)}$.