Lipeomorphic equivalence for p-adic analytic functions: a comparison between complex and p-adic dynamics

TL;DR

This paper investigates lipeomorphic equivalence of p-adic analytic functions tangent to the identity, comparing it with complex dynamics, and establishes optimal Holder estimates at zero.

Contribution

It determines conditions for lipeomorphic conjugacy of p-adic functions tangent to the identity and compares these results with complex and real cases.

Findings

Existence of lipeomorphic conjugacies under certain conditions

Optimal Holder estimates at zero for these conjugacies

Comparison between p-adic and complex dynamics results

Abstract

Let K be a p-adic field, and suppose that f and g are germs of analytic functions on K which are tangent to the identity at 0. It is known that f and g are homeomorphically equivalent, meaning there is an invertible germ h conjugating f to g. In this paper, we determine whether there exists such h which are lipeomorphisms, and moreover find the best possible Holder estimate at 0. Our results have striking complex and real counterparts.