Extension of formal conjugations between diffeomorphisms

TL;DR

This paper investigates the formal conjugacy properties of complex analytic diffeomorphisms near fixed points, revealing conditions under which formal conjugations cannot extend transversally along the fixed points set.

Contribution

It demonstrates that certain formal conjugations do not extend to the fixed points set and identifies geometric configurations that prevent such extensions in unfoldings of tangent to identity diffeomorphisms.

Findings

Existence of formal conjugations that do not extend transversally

Identification of geometric obstructions to extension

Analysis focused on unfoldings of tangent to identity diffeomorphisms

Abstract

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ${\mathbb C}^{n}$. More precisely, we are interested on the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms $\phi, \eta$ such that every formal conjugation $\hat{\sigma}$ (i.e. $\eta \circ \hat{\sigma} = \hat{\sigma} \circ \phi$) does not extend to the fixed points set $Fix (\phi)$ of $\phi$, meaning that it is not transversally formal (or semi-convergent) along $Fix (\phi)$. We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.