Affine mappings of translation surfaces: Geometry and arithmetic.

*(English)*Zbl 0965.30019The main result of this paper is an explicit general formula for the quadratic constant, \(c(S)\), where \(S\) is a Veech surface. The proof is based on the counting of vectors in the orbit of a lattice which is reduced to a counting of horocycles in the hyperbolic plane. The constant \(c(S)\) determines the main term of the asymptotics of the lengths of closed geodesics on \(S\). In the concluding section of this paper the authors show that \(c(S)\), when multiplied by \(\Pi\text{(Area}(S))\), is an algebraic number.

Reviewer: A.P.Stone (Albuquerque)

##### MSC:

30F60 | Teichmüller theory for Riemann surfaces |

37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |

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

30F30 | Differentials on Riemann surfaces |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

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\textit{E. Gutkin} and \textit{C. Judge}, Duke Math. J. 103, No. 2, 191--213 (2000; Zbl 0965.30019)

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