Reversing symmetry group of and matrices with connections to cat maps and trace maps

Abstract

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.