Half-Iterates and Delta Conjectures

TL;DR

This paper compares two algorithms for solving Abel's equation, highlighting their differences in efficiency and the characterization of the principal solution, and aims to fill a knowledge gap regarding their intrinsic properties.

Contribution

It provides a detailed analysis of the contrasting algorithms and seeks to determine the exact value of the correction term delta in EJ's solution.

Findings

EJ algorithm is faster and more efficient.

ML evaluates a limit related to the principal solution.

The paper aims to identify the exact correction term delta.

Abstract

The vivid contrast between two competing algorithms for solving Abel's equation $g(\theta(x)) = g(x) + 1$, given $\theta(x)$, is easily sketched. EJ is faster and more efficient, but ML evaluates a limit characterizing the principal solution $g(x)$ directly. EJ finds $g(x)+\delta$, where $\delta$ is possibly nonzero but independent of $x$. If we were to know an exact expression for $\delta$, then the "intrinsicality" of ML would be subsumed by EJ. Filling this gap in our knowledge is the aim of this paper.