Dynamics of Rational Surface Automorphisms: Rotation Domains

TL;DR

This paper studies rational surface automorphisms with positive entropy, constructing examples with rotation domains containing fixed points, and introduces a global linearization model for these complex structures.

Contribution

It constructs a new example of rational surface automorphism with a rotation domain containing fixed points and develops a global linearization framework for such domains.

Findings

Existence of rotation domains with fixed points in rational surface automorphisms.

Construction of a global linearization model for these rotation domains.

Demonstration that such Fatou components cannot be embedded into complex Euclidean space.

Abstract

We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model.