The equivalence problem in analytic dynamics for $1$-resonance

TL;DR

This paper investigates the conjugacy problem in analytic dynamics near singular points, focusing on cases where normalizing transformations are k-summable, and explores the associated moduli spaces and geometric obstructions.

Contribution

It introduces a framework for understanding singularities with k-summable normalizations and analyzes their unfoldings to reveal geometric obstructions and moduli spaces.

Findings

Normalizing transformations can be k-summable for certain singularities.

Unfoldings of these singularities help understand convergence obstructions.

Moduli spaces are constructed for generic parameter families.

Abstract

When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in the neighborhood of a singular point? A way to answer is to use normal forms. But there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper we discuss the case of singularities for which the normalizing transformation is $k$-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singularities themselves, and the geometric obstructions to convergence of the normalizing transformations. We also present some moduli spaces for generic $k$-parameter families unfolding such singularities.