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

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.