Reversing symmetry group of Gl ( 2 , Z ) and PGl ( 2 , Z ) matrices with connections to cat maps and trace maps
Reversing symmetry group of Gl ( 2 , Z ) and PGl ( 2 , Z ) matrices with connections to cat maps and trace maps
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Gl(2,Z)和PGl(2,Z)矩阵的反转对称群与猫图和跟踪图的连接
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发表时间：
1997
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通讯作者：
J. Roberts
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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了一个群，称为反向对称群R，其对称形成了R的正规子群S。我们给出了与整数矩阵群Gl（2，Z）和PGl（2，Z）相关的动力系统中R（从而S）的完整特征。为此，我们使用数论中众所周知的方法，如Dirichlet的单位定理的二次域和Gaussel的结果的等价的整数二次型，并采用代数结构的模群PSl（2，Z）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）的跟踪地图，保持的Sparke-Vogt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 R with the symmetries forming a normal subgroup S of R. We give a complete characterization of R (and hence S) in the dynamical systems associated with the groups of integral matrices Gl(2,Z) and PGl(2,Z). To do this, we use well known methods of number theory, such as Dirichlet’s unit theorem for quadratic fields and Gauß’ results on the equivalence of integer quadratic forms, and employ the algebraic structure of the modular group PSl(2,Z) 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.