Planar structures and billiards in rational polygons: the Veech alternative

Contents §1. Introduction §2. Definitions and preliminary information 2.1. Planar structures 2.2. Billiards in rational polygons 2.3. Interval exchange transformations 2.4. Elementary planar structures §3. The Veech alternative 3.1. The stabilizer of a planar structure 3.2. Veech's theorem 3.3. Masur's lemma 3.4. Pencils of periodic trajectories §4. Examples of Veech's theorem 4.1. Planar structures on a torus 4.2. Planar structures with one singular point 4.3. Non-lattice stabilizers §5. Covers of planar structures 5.1. Finiteness of the number of covers 5.2. Relation with stabilizers §6. Another proof of Veech's theorem 6.1. Property A 6.2.