Local analytic conjugacy of resonant analytic mappings in two variables, in the non-archimedean setting

TL;DR

This paper investigates the local analytic conjugacy of two-dimensional invertible mappings over non-archimedean fields, establishing conditions under which formal equivalence implies analytic equivalence based on eigenvalue properties.

Contribution

It provides a criterion for analytic conjugacy of resonant analytic mappings in two variables over non-archimedean fields, linking formal and analytic equivalence under specific eigenvalue conditions.

Findings

Formal and analytic conjugacy are equivalent under given eigenvalue conditions.

Analytic conjugacy is characterized by formal conjugacy for certain resonant mappings.

Eigenvalue conditions determine the equivalence of mappings in the non-archimedean setting.

Abstract

In this note, we consider locally invertible analytic mappings in two dimensions, with coefficients in a non-archimedean field. Suppose such a map has a Jacobian with eigenvalues $\lambda_1$ and $\lambda_2$ so that $|\lambda_1|>1$ and $\lambda_2$ is a positive power of $\lambda_1$, or that $\lambda_1=1$ and $|\lambda_2|\neq 1$. We prove that two formal maps with eigenvalues satisfying either of these conditions are analytically equivalent if and only if they are formally equivalent.