On the complex dynamics of birational surface maps defined over number fields

TL;DR

This paper proves that birational surface maps over number fields with high dynamical degree satisfy energy conditions, leading to well-behaved complex dynamics and the existence of a canonical height function.

Contribution

It establishes that such maps automatically meet the Bedford-Diller energy condition after conjugacy, ensuring well-behaved dynamics and defining a canonical height.

Findings

Maps satisfy Bedford-Diller energy condition after conjugacy

Existence of a well-defined canonical height function

Dynamics are well-behaved under these conditions

Abstract

We show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford-Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well-behaved. We also show that there is a well-defined canonical height function.