Julia set describes quantum tunnelling in the presence of chaos

Abstract

We find that characteristics of quantum tunneling in the presence of chaos\ncan be regarded as a manifestation of the Julia set of the complex dynamical\nsystem. Several numerical evidences for the standard map together with a\nrigorous statement for the H\\'enon map are presented demonstrating that the\ncomplex classical paths which contribute to the semiclassical propagator are\ndense in the Julia set. Chaotic tunneling can thus be characterized by the\ntransitivity of the dynamics and high density of the trajectories on the Julia\nset.\n