Formally-Reversible Maps of C^2

TL;DR

This paper characterizes the generic reversible elements in the group of formally-invertible pairs of formal power series in two variables, describing their structure, conjugacy classes, and factorization properties.

Contribution

It provides a complete description of generic reversible elements in the group of formal power series in two variables, including explicit sequences and conjugacy classifications.

Findings

All generic reversibles are conjugate to elements in two explicit sequences.

Each reversible element can be expressed as a product of two finite even order elements.

Any product of reversibles can be reduced to a product of five reversibles.

Abstract

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. This paper is about reversibles in the group $G$ of formally-invertible pairs of formal power series in two variables, with complex coefficients. The main result is a description of the generic reversible elements of $G$. We list two explicit sequences of reversibles which between them represent all the conjugacy classes of such reversibles. We show that each such element is reversible by some element of finite order, and hence is the product of two elements of finite even order. Those elements that may be reversed by an involution are called {\em strongly reversible}. We also characterise these. We draw some conclusions about generic reversibles in the group of biholomorphic germs in two variables, and about the factorization of formal maps as products of reversibles. Specifically, each product of reversibles reduces to the product of five.