Dynamical compactifications of C^2

TL;DR

This paper develops methods for compactifying polynomial maps of C^2, analyzes their degree growth, and characterizes Green functions and normal forms depending on topological degree.

Contribution

It introduces new dynamical compactifications for polynomial maps of C^2 and establishes properties of degree growth and Green functions based on topological degree.

Findings

Degree growth sequence satisfies a linear integral recursion.

Green function is well behaved for maps of low topological degree.

Normal forms are provided for maps of maximum topological degree.

Abstract

We find good dynamical compactifications for arbitrary polynomial mappings of C^2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula. For maps of low topological degree we prove that the Green function is well behaved. For maps of maximum topological degree, we give normal forms.