正八边形中直线轨迹的符号编码
Symbolic coding for linear trajectories in the regular octagon
摘要：
我们考虑一个符号编码的线性轨迹在正八边形的相对两侧确定（更一般地说，在定期2n-xm）。每一个无限的轨迹给出了一个切割序列对应的序列的侧面击中。我们给出了一个明确的特征，这些切割序列。正方形的切割序列是研究得很好的Sturmian序列，可以根据斜坡的连分数扩展进行分析。我们介绍了一个类似的连分式算法，我们使用它来连接切割序列的轨迹与其斜率。我们的连续分数扩展的斜率给出了一个显式序列的替代操作，生成切割序列的轨迹与该斜率。我们的算法可以理解的Veech组的元素的八边形平移表面的重整化。
英文摘要：
We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides hit. We give an explicit characterization of these cutting sequences. The cutting sequences for the square are the well studied Sturmian sequences which can be analyzed in terms of the continued fraction expansion of the slope. We introduce an analogous continued fraction algorithm which we use to connect the cutting sequence of a trajectory with its slope. Our continued fraction expansion of the slope gives an explicit sequence of substitution operations which generate the cutting sequences of trajectories with that slope. Our algorithm can be understood in terms of renormalization of the octagon translation surface by elements of the Veech group.
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