Fractional Iteration of Series and Transseries

TL;DR

This paper develops a framework for fractional compositional iteration of transseries, extending classical iteration concepts to transseries of exponentiality zero, and explores their structural properties and classifications.

Contribution

It introduces a family of fractional iterates for transseries of exponentiality zero based on Abel's Equation and analyzes their support set properties.

Findings

Existence of fractional iterates $T^{[s]}$ for large positive transseries of exponentiality 0

Classification of transseries into shallow, moderate, and deep with distinct iteration properties

Identification of support set sharing among fractional iterates

Abstract

We investigate compositional iteration of fractional order for transseries. For any large positive transseries $T$ of exponentiality 0, there is a family $T^{[s]}$ indexed by real numbers $s$ corresponding to teration of order $s$. It is based on Abel's Equation. We also investigate the question of whether there is a family $T^{[s]}$ all sharing a single support set. A subset of the transseries of exponentiality 0 is divided into three classes ("shallow", "moderate" and "deep") with different properties related to fractional iteration.