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

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 $\mathrm{\mid }{\lambda }_1\mathrm{\mid }>1$ and ${\lambda }_2$ is a positive power of ${\lambda }_1$, or that ${\lambda }_1=1$ and $\mathrm{\mid }{\lambda }_2\mathrm{\mid }\mathrm{\ne }1$. We prove that two formal maps with eigenvalues satisfying either of these conditions are analytically equivalent if and only if they are formally equivalent.