Julia set describes quantum tunneling in the presence of chaos
  We find that characteristics of quantum tunneling in the presence of chaos can be regarded as a manifestation of the Julia set of the complex dynamical system. Several numerical evidences for the standard map together with a rigorous statement for the H\'enon map are presented demonstrating that the complex classical paths which contribute to the semiclassical propagator are dense in the Julia set. Chaotic tunneling can thus be characterized by the transitivity of the dynamics and high density of the trajectories on the Julia set. 
A. Shudo, Y. ishii and K.S. Ikeda