### A $p$-adic approach to local analytic dynamics: analytic conjugacy of analytic maps tangent to the identity

In this note, we consider the question of local analytic equivalence of analytic functions which fix the origin and are tangent to the identity. All mappings and equivalences are considered in the non-archimedean context e.g. all norms can be considered $p$-adic norms. We show that any two mappings $f$ and $g$ which are formally equivalent are also analytically equivalent. We consider the related questions of roots and centralizers for analytic mappings. In this setting, anything which can be done formally...