The Dynamics of Trace Maps

Abstract

Trace maps are 3-dimensional (3D) mappings derived from transfer matrix approaches to various physical processes that display quasiperiodicity in space or time. A subset of trace maps, to be considered here, possess one (and the same) constant of motion and so induce dynamics on the 2D level sets of this integral. When motivated by a particular physical problem, these trace maps are often studied in a particular regime of the phase space and with particular initial conditions. Here we take a global interest in them, motivated more by them being interesting dynamical systems. We show that, particularly with regard to the structure and location of their periodic orbits, these mappings are closely related to 2D Hamiltonian (i.e., area-preserving) mappings. We highlight a particular level set where the dynamics is exactly solvable and closely related to that of a hyperbolic toral automorphism or ‘cat map’. Finally, we show how time-reversal symmetry in such trace maps can be identified from a related property of matrices belonging to PGl(2, ℤ).