Translation Surfaces

Cover Title page Copyright Contents Preface Chapter 1. Introduction 1.1. The square torus 1.2. The space of tori 1.3. Dynamics 1.4. Counting Chapter 2. Three Definitions 2.1. Polygons 2.2. Geometric structures 2.3. Holomorphic 1-forms 2.4. Saddle connections and cylinders 2.5. Equivalences 2.6. Quadratic differentials Chapter 3. Moduli Spaces of Translation Surfaces 3.1. Teichmüller and moduli spaces 3.2. Strata of translation surfaces 3.3. Coordinates on components of strata 3.4. Measures 3.5. Components of strata 3.6. Delaunay triangulations 3.7. Finiteness of the MSV measure Chapter 4. Dynamical Systems and Ergodic Theory 4.1. Topological dynamics 4.2. Ergodic theory 4.3. Low genus examples 4.4. Number of ergodic measures 4.5. Interval exchange transformations Chapter 5. Renormalization 5.1. A criterion for unique ergodicity 5.2. Ergodicity of the Teichmüller flow 5.3. Geodesics, horocycles, and mixing 5.4. Quantitative renormalization and ergodicity 5.5. Nonunique ergodicity and Hausdorff dimension Chapter 6. Counting and Equidistribution 6.1. Lattice point counting 6.2. Saddle connections and holonomy vectors 6.3. Siegel-Veech transforms 6.4. From counting to dynamics 6.5. Equidistribution of circles 6.6. Further counting results 6.7. Counting and volumes Chapter 7. Lattice Surfaces 7.1. Affine diffeomorphisms and Veech groups 7.2. Examples 7.3. Cusps, cylinders, and shears 7.4. Optimal dynamics 7.5. Homogeneous dynamics 7.6. Characterizations of lattice surfaces 7.7. Classifying lattice surfaces 7.8. Square-tiled surfaces Conclusion Bibliography Index Back Cover