Half-Iterates and Delta Conjectures

TL;DR

The paper investigates Abel's equation $g(\theta(x))=g(x)+1$ via two numerical schemes, EJ and ML, highlighting that EJ yields a constant offset $\delta$ so that $g(x)\approx g(x)+\delta$, while ML directly targets the principal solution through a limiting procedure. It presents multiple explicit computations for families $\theta(x)$, documents a recurring pattern where $\delta$ depends on local parameters, and proposes generalized delta formulas across cubic and higher-order forms, including for certain transcendental cases. Addenda I–III introduce new recurrences, derive asymptotic representations for generalized half-iterates, and conjecture broad relations tying $\delta$ to parameters like $\tau$, $\sigma$, and $\rho$, suggesting a unified framework for predicting half-iterate corrections across a wide class of analytic $\theta$. These results deepen understanding of the intrinsic vs. extrinsic characterizations of half-iterates and offer practical guidance for computing principal solutions and their refinements in functional equations.

Abstract

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

Figures (2)

• Figure 1: Blue curve is $10\theta_{13}(x)$, scaled for visibility; green curve is ML-based $\tilde{g}_{13}(x)$. Lower left-hand corner is $(0,1)$. Distance between vertical notches is eight times that for horizontal.
• Figure 2: Half-iterate of $x\left( 1-x^{2}\right)$ in orange; $x\left( 1-x^{2}\right)$ in blue; diagonal line in red. Lower left-hand corner is $(0,1/10)$.