标题： 一种广义的多维连分数族：TRIP 映射 显示英文标题

标题： A Generalized Family of Multidimensional Continued Fractions: TRIP Maps

摘要： 最著名的多维连分数，包括Mönkemeyer映射和三角形映射，都是通过反复细分三角形生成的。本文通过在每次细分前后对这些三角形的顶点进行排列，构造了一类多维连分数。通过组合该类中的映射，我们得到了一个更大的多维连分数类。这些包括Brun、Parry-Daniels和Güting的算法。我们给出了多维连分数将序列与唯一点相关联的条件，这使我们能够确定当相应的多维连分数的周期性对应于实数对为同一数域中的三次无理数时的情况。 显示英文摘要

摘要： Most well-known multidimensional continued fractions, including the M\"{o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and G\"{u}ting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.