天使的楼梯，Sturmian序列，以及在相似曲面上的轨迹
Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces
摘要：
一个相似曲面可以从多边形中组装出来，通过相似性来识别它们的边，这是平移和缩放的组合。我们考虑一类单参数亏格2相似曲面族上的线性轨线。这些曲面上的轨迹的闭合总是具有豪斯多夫维数1，并且包含具有康托横截面的闭合环或层。轨迹的切割序列要么最终是周期性的，要么最终是Sturmian的。虽然没有两个这些表面是affirmative等效的，他们的线性轨迹可以直接相关的平方环面上，从而彼此，通过显式函数。我们还简要地检查了两个相关的家庭的表面，并表明上述行为可以混合，例如，封闭的线性轨迹可以包含一个封闭的循环和层压。
英文摘要：
A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
