2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146110We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.to appear in Duke Mathematical Journal. Some minor corrections made, typos fixed, some new references addedDynamical SystemsGeometric Topology37E35; 37F25text