B\"ottcher coordinates

TL;DR

This paper generalizes Böttcher's theorem, which describes how certain complex functions near a superattracting fixed point can be simplified via an analytic change of variables, to the setting of multiple complex variables.

Contribution

It extends Böttcher's conjugacy result from one complex variable to several complex variables, broadening the theorem's applicability.

Findings

Established a conjugacy to a monomial map in multiple variables

Generalized Böttcher coordinates to higher dimensions

Provided conditions for the existence of such conjugacies

Abstract

A well-known theorem of B\"ottcher asserts that an analytic germ f:(C,0)->(C,0) which has a superattracting fixed point at 0, more precisely of the form f(z) = az^k + o(z^k) for some a in C^*, is analytically conjugate to z->az^k by an analytic germ phi:(C,0)->(C,0) which is tangent to the identity at 0. In this article, we generalize this result to analytic maps of several complex variables.