Rational billiards and flat structures

Abstract

1 Polygonal billiards, rational billiards 1 1.1 Polygonal billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Examples: a pair of ellastic point-masses on a segment and a triple of point-masses on a circle . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Unfolding billiard trajectories, rational polygons . . . . . . . . . . . . 3 1.4 Example: billiard in the unit square . . . . . . . . . . . . . . . . . . . 4 1.5 Rational billiard determines a flat surface . . . . . . . . . . . . . . . . 6 1.6 Minimality of the billiard flow in rational polygons . . . . . . . . . . 8 1.7 Rational billiards and interval exchange maps . . . . . . . . . . . . . 11 1.8 Flat metrics and quadratic differentials . . . . . . . . . . . . . . . . . 12