Chaos attracts tunneling trajectories: A universal mechanism of chaotic tunneling

It is shown that stable and unstable manifolds in the complex plane control dynamical tunneling in mixed systems. Some hypotheses based upon numerical observations combined with recent results in the theory of complex dynamical systems (Bedford and Smillie 1991, 1992) reveal that the stable manifolds in complex space are densely distributed in the set of tunneling orbits which asymptotically contribute to the semiclassical propagator. It is indeed verified that the tunneling amplitude out of the KAM region is well reproduced by the orbits attracted by unstable periodic orbits in the real phase space.