Orateur
Description
Given a tangent to the identity germ of holomorphic diffeomorphism $\phi$, we consider the map $P \mapsto \phi_{P}$ that associates to any fixed point $P$ of $\phi$ near the origin the germ $\phi_{P}$ of $\phi$ at $P$. Such germs are not in general tangent to the identity. Given the infinitesimal generator $X$ of $\phi$, a formal vector field, it is natural to ask whether we can define ``infinitesimal generators" $X_{P}$ of $\phi_{P}$ for $P \in \mathrm{Fix} (\phi)$ and if the dependence of $X_{P}$ on $P \in \mathrm{Fix} (\phi)$ is analytic. In other works, we are asking whether $X$ is formally transversal along $\mathrm{Fix} (\phi)$.
We introduce a weak and a strong concept of formal transversality for $X$. On the one hand, we will see that $X$ is always formally transversal along $\mathrm{Fix} (\phi)$ in the weak sense. On the other hand, there are examples where $X$ is not formally transversal along $\mathrm{Fix} (\phi)$ in the strong sense. We will discuss the gap between the weak and the strong concepts and provide a characterization of strong formal transversality. This is a joint work with Rudy Rosas.
