Escaping orbits in trace maps

We study the finitely-generated group A of invertible polynomial mappings from 3 to itself (or 3 to itself) which preserve the Fricke-Vogt invariant I(x, y, z) = x2 + y2 + z2 − 2xyz − 1. Using properties of suitably-chosen generators, we give a necessary condition and sufficient conditions for infinite order elements of A to have an unbounded orbit escaping to infinity in forward or backward time. Our main motivation for this study is that A includes the so-called trace maps derived from transfer matrix approaches to various physical processes displaying non-periodicity in space or time. As shown previously, characterising escaping orbits leads to various conclusions for the physical model and vice versa (e.g. electronic properties of ID quasicrystals). Our results generalize in a simple constructive way those previously proved for Fibonacci-type trace maps.