平移曲面与平面共振集的丢番图逼近
Diophantine approximations for translation surfaces and planar resonant sets
AI总结：
本文利用平移曲面产生的平面共振集，研究 Teichmüller 测地线有界性、规定速率偏移及有理台球快速回复方向的 Hausdorff 维数，建立了良好逼近与不良逼近方向的二分法及维数估计。
摘要：
我们考虑平移曲面层中的 Teichmüller 测地线。我们证明了生成在层的某个紧致部分中有界的测地线的参数集合的 Hausdorff 维数的下界和上界。然后我们计算了那些生成以规定速率向无穷远偏移的测地线的参数的维数。最后我们计算了有理台球中具有快速回复性的方向集合的维数，这对应于 Jarník 和 Besicovich 经典结果的一个动力系统版本。我们的主要工具是由给定平移曲面产生的平面共振集，即其鞍形连接或闭测地线的可数方向集合，按长度进行过滤。在一个抽象框架中，并假设一般平面共振集上具有特定的度量性质，我们证明了被共振集中的方向良好逼近的方向集合的 Hausdorff 测度的一个二分法，并给出了被不良逼近的方向集合的维数的一个估计。然后我们证明由平移曲面产生的共振集满足所需的度量性质。
英文摘要：
We consider Teichmüller geodesics in strata of translation surfaces. We prove lower and upper bounds for the Hausdorff dimension of the set of parameters generating a geodesic bounded in some compact part of the stratum. Then we compute the dimension of those parameters generating geodesics that make excursions to infinity at a prescribed rate. Finally we compute the dimension of the set of directions in a rational billiard having fast recurrence, which corresponds to a dynamical version of a classical result of Jarník and Besicovich. Our main tool are planar resonant sets arising from a given translation surface, that is the countable set of directions of its saddle connections or of its closed geodesics, filtered according to length. In an abstract setting, and assuming specific metric properties on a general planar resonant set, we prove a dichotomy for the Hausdorff measure of the set of directions which are well approximable by directions in the resonant set, and we give an estimate on the dimension of the set of badly approximable directions. Then we prove that the resonant sets arising from a translation surface satisfy the required metric properties.
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