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Veech surfaces and complete periodicity in genus two. (English) Zbl 1073.37032

A translation surface is a closed Riemannian manifold \(M\) with an Abelian differential \(\omega\) on \(M\) satisfying that away from the zeros of \(\omega\) the charts can be chosen such that the change of charts are translations. Consider the geodesics on this surface. A direction \(v\) is said to be completely periodic if every trajectory in direction \(v\) is either closed or begins and ends in a singularity. The surface \((M,\omega )\) is called completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of \(\text{SL}(2,\mathbb{R})\) on the space of translation surfaces. A translation surface having a lattice stabilizer under this action is called Veech surface.
According to W. Veech [Invent. Math. 97, 553–583 (1989; Zbl 0676.32006)], every Veech surface is completely periodic, but the converse need not be true. In this paper, translation surfaces of genus two are investigated. Then either \(\omega\) has two single zeros or \(\omega\) has one double zero. Conditions implying the surface to be a Veech surface are presented. The main tool in the proofs is a method developped by R. Kenyon and J. Smillie [Comment. Math. Helv. 75, 65–108 (2000; Zbl 0967.37019)].
Reviewer: Peter Raith (Wien)

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37E05 Dynamical systems involving maps of the interval
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
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References:

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