Topology and geometry of the general composition of formal power series -- towards Fr\'echet-Lie group-like formalism

TL;DR

This paper investigates the topological and geometric properties of formal power series under composition, establishing a Fréchet-Lie group structure and analyzing the superposition operator's smoothness and invertibility.

Contribution

It introduces a Fréchet-Lie group framework for formal power series and characterizes conditions for invertibility under composition.

Findings

Superposition operator is continuous and smooth in the Fréchet topology.

Characterization of invertibility conditions for formal power series.

Establishment of a Fréchet-Lie group structure on nonunit formal power series.

Abstract

In this article, we study the properties of the autonomous superposition operator on the space of formal power series, including those with nonzero constant term. We prove its continuity and smoothness with respect to the topology of pointwise convergence and a natural Fr\'echet manifold structure. A necessary and sufficient condition for the left composition inverse of a formal power series to exist is provided. We also present some properties of the Fr\'echet-Lie group structures on the set of nonunit formal power series.