B\"ottcher coordinates at superattracting fixed points of holomorphic skew products

TL;DR

This paper constructs Böttcher coordinates for holomorphic skew products with superattracting fixed points, enabling conjugation to monomial maps under certain weight conditions, thus extending classical one-variable results to a two-variable setting.

Contribution

It introduces a method to construct Böttcher coordinates for a class of holomorphic skew products in two variables with superattracting fixed points, under specific weight conditions.

Findings

Böttcher coordinates exist under certain weight conditions.

Coordinates conjugate the map to a monomial map.

Construction applies to open sets containing the fixed point.

Abstract

Let $f : (\mathbb{C}^2, 0) \to (\mathbb{C}^2, 0)$ be a germ of holomorphic skew product with a superattracting fixed point at the origin. If it has a suitable weight, then we can construct a B\"ottcher coordinate which conjugates $f$ to the associated monomial map. This B\"ottcher coordinate is defined on an open set whose interior or boundary contains the origin.