Reversible biholomorphic germs

TL;DR

This paper characterizes reversible elements and their reversers in the group of biholomorphic germs at the origin in one complex variable, providing a complete description of solutions to a specific conjugation equation.

Contribution

It offers a complete characterization of reversible biholomorphic germs and describes all reversers, advancing understanding of symmetry properties in complex dynamics.

Findings

Characterization of all reversible elements in the group of biholomorphic germs.

Description of the set of reversers for each reversible element.

Solution to the conjugation equation for holomorphic functions near the origin.

Abstract

Let $G$ be a group. We say that an element $f\in G$ is {\em reversible in} $G$ if it is conjugate to its inverse, i.e. there exists $g\in G$ such that $g^{-1}fg=f^{-1}$. We denote the set of reversible elements by $R(G)$. For $f\in G$, we denote by $R_f(G)$ the set (possibly empty) of {\em reversers} of $f$, i.e. the set of $g\in G$ such that $g^{-1}fg=f^{-1}$. We characterise the elements of $R(G)$ and describe each $R_f(G)$, where $G$ is the the group of biholomorphic germs in one complex variable. That is, we determine all solutions to the equation $ f\circ g\circ f = g$, in which $f$ and $g$ are holomorphic functions on some neighbourhood of the origin, with $f(0)=g(0)=0$ and $f'(0)\not=0\not=g'(0)$.