Long time semiclassical propagation in a perturbed cat map

Abstract We recently presented leading-order semiclassical expressions for the matrix elements of the evolution operator between wave functions constructed in the vicinity of unstable periodic orbits. By employing a novel canonical transformation, these expressions remain valid for arbitrarily long times. In this work, we analyse this transformation within the framework of a perturbed cat map and compare it with the analytical method of hyperbolic normal forms, revealing a connection between them. Based on these insights, we hypothesise that by introducing a small number of transverse excitations to the wave functions, the semiclassical expressions can also incorporate the next-order correction in ℏ . To validate this hypothesis, we test our approach in a quantum version of the perturbed cat map.