Time：16:00-18:00, Tuesday, August 18 2026
Venue：E14-317
Speaker：Xiao Li，Centre de Mathématiques Laurent Schwartz, École Polytechnique
Title：Energy Conditions for Birational Maps of Surfaces
Abstract：In this talk, I will present the work of Jonsson and Reschke on the complex dynamics of birational surface maps. Bedford and Diller introduced an energy condition for birational self-maps of smooth complex projective surfaces whose first dynamical degree is greater than one. Under this condition, the canonical invariant currents admit a well-defined intersection, giving rise to a natural invariant probability measure with strong ergodic properties.
Jonsson and Reschke proved that, when both the surface and the map are defined over a number field, the Bedford–Diller energy condition is automatically satisfied after passing to a suitable birational model. Combined with results of Bedford and Diller, Dujardin, and Diller, Dujardin, and Guedj, this implies that the resulting equilibrium measure is mixing, hyperbolic, and of maximal entropy. Moreover, the saddle periodic points equidistribute with respect to this measure. I will also discuss the arithmetic ideas underlying the proof, particularly the role of canonical heights, and conclude with some generalizations of these results.
