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We investigate in what regime of the frequency the new tunneling mechanism using unstable and stable manifolds of dividing saddle (Takahashi and Ikeda 2008 J. Phys. A: Math. Theor. 41 095101; Takahashi and Ikeda 2009 Phys. Rev. A 79 052114) works as the leading mechanism of multi-dimensional barrier tunneling. In the large and small limits of the frequency, the tunneling rate is well evaluated based on the instanton picture. It is in the intermediate frequency range that the new mechanism dominates the tunneling process. The tunneling rate takes the maximum value in this intermediate range.

The tunneling phenomenon in non-integrable systems is studied in the framework of complex semiclassical theory. Complex trajectories which dominate tunneling in the presence of chaos (chaotic tunneling) are investigated numerically for several quantum maps. The discovery of a characteristic structure in the initial value representation of tunneling trajectories, named the Laputa chain, is reviewed, and it is shown how trajectories starting from Laputa chains make the dominant contribution to the semiclassical calculation of the wavefunction in the chaotic regime. This supports the argument that Laputa chains play an important role in the fully complex-domain semiclassical description of chaotic tunneling. Further, numerical analysis shows that the Laputa chain has distinct asymptotic properties in the long time limit. In particular, it is shown that the imaginary action along the trajectories starting from the Laputa chain, which determines the contribution to the tunneling probability, tends to converge absolutely in the asymptotic limit. On the basis of these features, we propose an empirical definition of the Laputa chain which can provide a basis for further mathematical development. Moreover, a connection is pointed out between the asymptotic structure of Laputa chains and Julia sets manifest in asymptotic dynamics of complex maps. Based on these results, we make the conjecture that Julia sets play a fundamental role in the complex semiclassical dynamical theory of tunneling in non-integrable systems.