扩展模群的法雷图与有理不动点

IF 0.7 Q2 MATHEMATICS

Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics Pub Date : 2022-12-30 DOI:10.31801/cfsuasmas.1089480

Bilal Demir, Mustafa Karataş

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引用次数: 0

摘要

矩阵的不动点在科学和数学的各个领域都有许多应用。扩展模群ΓΓ是一组2×22×2矩阵，具有整数项和行列式±1±1。扩展模群、连分式和Farey图之间有很强的联系。Farey图是一个顶点集为^Q=Qõ｛∞｝Q^=Qõ{∞}的图。在这项研究中，我们考虑了ΓΓ中固定有理数的元素。对于给定有理数，我们使用它的Farey邻居来获得$\overline｛\Gamma｝$中元素的矩阵表示，该矩阵表示固定了给定有理数。然后，我们使用Farey图和连分式之间的关系，用生成器将这些元素表示为单词。最后，我们给出了这些字的新的块简化形式，所有块都有Fibonacci数条目。

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Farey graph and rational fixed points of the extended modular group

Fixed points of matrices have many applications in various areas of science and mathematics. Extended modular group ¯¯¯¯ΓΓ¯ is the group of 2×22×2 matrices with integer entries and determinant ±1±1. There are strong connections between extended modular group, continued fractions and Farey graph. Farey graph is a graph with vertex set ^Q=Q∪{∞}Q^=Q∪{∞}. In this study, we consider the elements in ¯¯¯¯ΓΓ¯ that fix rationals. For a given rational number, we use its Farey neighbours to obtain the matrix representation of the element in $\overline{\Gamma}$ that fixes the given rational. Then we express such elements as words in terms of generators using the relations between the Farey graph and continued fractions. Finally we give the new block reduced form of these words which all blocks have Fibonacci numbers entries.

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Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics MATHEMATICS-

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