Reversing symmetry group of and matrices with connections to cat maps and trace maps
Reversing symmetry group of and matrices with connections to cat maps and trace maps
复制标题
与猫图和迹图连接的矩阵的反转对称群
DOI：
10.1088/0305-4470/30/5/020
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发表时间：
1997
期刊：
影响因子：
--
通讯作者：
J. Roberts
中科院分区：
文献类型：
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作者：
M. Baake;J. Roberts
动力系统可以同时具有对称性和时间反演对称性。这两种对称一起形成一个群，称为反向对称群，对称形成的正规子群。我们给出了一个完整的表征（因此）的动力系统与群的积分矩阵和。要做到这一点，我们使用众所周知的方法数论，如狄利克雷单位定理的二次领域和高斯的结果等价的整数二次形式，并采用代数结构的模块组作为一个自由的产品。我们展示了最近讨论的一些广义的反向对称群也很好地说明了当我们考虑仿射扩展这些矩阵群。我们的结果是适用于双曲型toral自同构（Anosov或猫映射），伪Anosov映射，和组的三维（3D）的迹映射，保持的Sparke-Vogt沃格特不变。
Dynamical systems can have both symmetries and time-reversing symmetries. Together these two types of symmetries form a group called the reversing symmetry group with the symmetries forming a normal subgroup of . We give a complete characterization of (and hence ) in the dynamical systems associated with the groups of integral matrices and . To do this, we use well known methods of number theory, such as Dirichlet's unit theorem for quadratic fields and Gaus' results on the equivalence of integer quadratic forms, and employ the algebraic structure of the modular group as a free product. We show how some recently discussed generalizations of the reversing symmetry group are also nicely illustrated when we consider affine extensions of these matrix groups. Our results are applicable to hyperbolic toral automorphisms (Anosov or cat maps), pseudo-Anosov maps, and the group of three-dimensional (3D) trace maps that preserve the Fricke - Vogt invariant.