Hyperbolic polarized dynamics, pairs of inverse maps and the Dirichlet property

TL;DR

This paper investigates the intersection properties of divisors in polarized dynamical systems on algebraic surfaces, establishing conditions for hyperbolic polarizations and analyzing the Dirichlet property in specific automorphisms.

Contribution

It introduces new geometric conditions for hyperbolic polarizations and provides examples of automorphisms on K3 surfaces lacking the Dirichlet property.

Findings

Necessary conditions for hyperbolic polarizations

Existence of automorphisms without the Dirichlet property on K3 surfaces

Analysis of divisor intersection properties

Abstract

We explore some intersection properties of divisors associated to polarized dynamical systems on algebraic surfaces. As a consequence, we obtain necessary geometric conditions for the existence of polarizations of hyperbolic type and exhibit compactified divisors associated to automorphisms on K3 surfaces that do not have the Dirichlet property as defined by Moriwaki.