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

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.

Added simplified argument for algebraically stable maps on the projective plane. To appear in Crelle's Journal. 19 pages