PAN, Huiping; SU, Weixu; NGUYEN, Duc-Manh

doi:10.1007/s00208-019-01897-2

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PAN, Huiping

SU, Weixu
Fudan University [Shanghai]

NGUYEN, Duc-Manh
Institut de Mathématiques de Bordeaux [IMB]

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Mathematische Annalen. 2020

Springer Verlag

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We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa's classifications of periodic points and of $\mathrm{GL}(2,\mathbb{R})$ orbit closures in hyperelliptic components, as well as a recent result of Eskin-Filip-Wright.< Leer menos

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Existence of closed geodesics through a regular point on translation surfaces

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