A construction of B\"{o}ttcher coordinates for holomorphic skew products

TL;DR

This paper constructs Böttcher coordinates for certain holomorphic skew products near a superattracting fixed point, showing they are conjugate to monomial maps under specific conditions, based on degree and Newton polygon.

Contribution

It provides a method to explicitly construct Böttcher coordinates for holomorphic skew products with superattracting fixed points, extending classical results to this class of maps.

Findings

Existence of conjugacy to monomial maps under specified conditions

Construction of Böttcher coordinates in this setting

Dependence on degree of p and Newton polygon of q

Abstract

Let $f(z,w)=(p(z),q(z,w))$ be a holomorphic skew product with a superattracting fixed point at the origin. Under one or two assumptions, we prove that $f$ is conjugate to a monomial map on an invariant open set whose closure contains the origin. The monomial map and the open set are determined by the degree of $p$ and the Newton polygon of $q$.