Mattias Jonsson: On the complex dynamics of birational surface maps defined over number fields.

Iterating a birational selfmap of the complex projective plane can lead to very interesting dynamics. Some special cases are reasonably well understood, such as the complex Henon maps, but the general picture is only known under certain hypotheses that may be hard to verify.

I will report on joint work with Paul Reschke, where we prove that if the map has rational coefficients (or more generally is defined over a number field) then the so-called Bedford-Diller energy condition is automatically satisfied, and the complex dynamics is well behaved. While our conclusion concerns complex dynamics, the main techniques come from the study of algebraic surfaces and of heights of algebraic numbers.