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

TL;DR

This paper explores local analytic dynamics over non-archimedean fields, demonstrating that formal equivalences imply analytic equivalences and that analytic vector fields generate flows, extending classical results to p-adic contexts.

Contribution

It establishes that formal equivalence implies analytic equivalence for p-adic analytic functions tangent to the identity and shows that analytic vector fields generate flows in this setting.

Findings

Formal and analytic equivalences coincide in the p-adic setting.

Analytic vector fields produce analytic flows.

Results extend classical local dynamics to non-archimedean fields.

Abstract

In this note, we will consider the question of local equivalence of analytic functions which fix the origin and are tangent to the identity, as well as the question of flows of analytic vector fields. 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, and we show that analytic vector fields generate analytic flows. We consider the related questions of roots and centralizers for analytic mappings. In this setting, anything which can be done formally can also be done analytically.