From the Introduction: Let \(\sigma\) be a holomorphic map defined near the origin of \({\mathbb C}^2\) with \(\sigma(0)=0\). We say that \(\sigma\) is {reversible} by an involution \(\tau\) (\(\tau^2=\text{ Id})\), \(\tau(0)=0\), if \(\sigma^{-1}=\tau\sigma\tau\). We say that \(\sigma\) is {holomorphically reversible} if \(\tau\) is additionally holomorphic, or is {anti-holomorphically reversible} if \(\tau\) is additionally anti-holomorphic.
In this paper we prove
Theorem 1.1. There exists a holomorphic map \(\sigma\) of \({\mathbb C}^2\) of the form \(\xi\to\lambda\xi+O(2),\eta\to\bar{\lambda}\eta+O(2)\), with \(\lambda\) not a root of unity and \(| \lambda| =1\), such that \(\sigma\) is reversible by an anti-holomorphic involution and by a formal holomorphic involution, and is however not reversible by any \({\mathcal C}^1\)-smooth involution of which the linear part is holomorphic. In particular, the \(\sigma\) is not reversible by any holomorphic involution.
For the entire collection see [Zbl 1051.32001].