Siegel Disks on Rational Surfaces

TL;DR

This paper constructs specific automorphisms on rational surfaces that have positive entropy and multiple Siegel disks, expanding understanding of complex dynamics on algebraic surfaces.

Contribution

It demonstrates the existence of rational surface automorphisms with prescribed numbers of Siegel disks and positive entropy, especially from quadratic birational maps.

Findings

Existence of rational surface automorphisms with Siegel disks and positive entropy.

Construction of automorphisms from quadratic birational maps fixing cubic curves.

Automorphisms with multiple Siegel disks and positive entropy.

Abstract

We show the existence of a rational surface automorphism of positive entropy with a given number of Siegel disks. Moreover, among automorphisms obtained from quadratic birational maps on the projective plane fixing irreducible cubic curves, we find out an automorphism of positive entropy with multiple Siegel disks.