Reversing symmetry group of Gl(2,Z) and PGl(2,Z) matrices with connections to cat maps and trace maps

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.