Planar structures and billiards in rational polygons: the Veech alternative.

*(English. Russian original)*Zbl 0897.58029
Russ. Math. Surv. 51, No. 5, 779-817 (1996); translation from Usp. Mat. Nauk 51, No. 5, 3-42 (1996).

This is an expository article presenting the background of the theory of so-called planar structures. Let \(M\) be a compact, connected, and orientable surface. A planar structure on \(M\) is formed by a maximal atlas \(\omega\) on \(M\setminus \{x_1,\ldots ,x_k\}\) (for some points \(x_1,\ldots ,x_k\) in \(M\)) such that all coordinate changing functions are shifts in \(\mathbb{R}^2\) and for each \(x_i\) there are a punctured neighborhood \(\dot U_i\) not containing \(x_j\), \(j\neq i\), and a map \(f_i:\dot U^i\to \dot V\), where \(\dot V\) is a punctured neighborhood of a point in \(\mathbb{R}^2\), which is a shift in the local coordinates from \(\omega\) and such that each point in \(\dot V\) has exactly \(m_i\) preimages. Planar structures are sometimes called quadratic differentials (in Teichmüller theory) or measured foliations. They appear naturally in studying billiards in rational polygons.

The paper consists of six sections: 1. Introduction, 2. Definitions and preliminary information, 3. The Veech alternative, 4. Examples of Veech’s theorem, 5. Covers of planar structures, 6. Another proof of Veech’s theorem. In particular, Section 2 contains connections to the theory of billiards and interval exchange transformations, a result on uniqueness of the planar structure on the torus and results on global properties of the geodesic flow. In Section 3, new proofs of Veech’s theorem on the stabilizer of a planar structure and Masur’s lemma are given. Several examples (including new ones) relating to Veech’s theorem are contained in Section 4. Results on the number of covers for a given planar structure and relations of covers with stabilizers are given in Section 5. The proof of Veech’s theorem in Section 6 is based on some geometric considerations.

The paper consists of six sections: 1. Introduction, 2. Definitions and preliminary information, 3. The Veech alternative, 4. Examples of Veech’s theorem, 5. Covers of planar structures, 6. Another proof of Veech’s theorem. In particular, Section 2 contains connections to the theory of billiards and interval exchange transformations, a result on uniqueness of the planar structure on the torus and results on global properties of the geodesic flow. In Section 3, new proofs of Veech’s theorem on the stabilizer of a planar structure and Masur’s lemma are given. Several examples (including new ones) relating to Veech’s theorem are contained in Section 4. Results on the number of covers for a given planar structure and relations of covers with stabilizers are given in Section 5. The proof of Veech’s theorem in Section 6 is based on some geometric considerations.

Reviewer: R.Srzednicki (Kraków)

##### MSC:

37A99 | Ergodic theory |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |

30F60 | Teichmüller theory for Riemann surfaces |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |