Reversible biholomorphic germs

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 $f'(0)\not=0\not=g'(0)$.