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In this note, we will consider the question of local equivalence of analyticfunctions which fix the origin and are tangent to the identity, as well as thequestion of flows of analytic vector fields. All mappings and equivalences areconsidered 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 formallyequivalent are also analytically equivalent, and we show that analytic vectorfields generate analytic flows. We consider the related questions of roots andcentralizers for analytic mappings. In this setting, anything which can be doneformally can also be done analytically.

A p-adic approach to local analytic dynamics: analytic flows and analytic maps tangent to the identity