Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy

TL;DR

This paper demonstrates that certain complex analytic unipotent germs of diffeomorphisms in multiple variables cannot be embedded into a flow through formal conjugacy, revealing limitations in the formal classification of such transformations.

Contribution

It provides the first examples of non-embeddable formal classes of unipotent germs in higher dimensions, using potential theory and linear functional operators.

Findings

Existence of non-embeddable unipotent germs in complex dimension greater than one.

Construction of examples within a geometrically defined family.

Application of potential theory to analyze linear operators associated with diffeomorphisms.

Abstract

The formal class of a germ of diffeomorphism $\phi$ is embeddable in a flow if $\phi$ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at $({\mathbb C}^{n},0)$ ($n>1$) whose formal class is non-embeddable. The examples are inside a family in which the non-embeddability is of geometrical type. The proof relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms via potential theory.