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Complex orbits connecting regular and chaotic regions are exponentially many. One can prove it using the fact that stable manifold for an arbitrary unstable periodic orbit is dense in the forward Julia set. It could further be proved that they (i) act as tunneling orbits when they stay in the torus region, and (ii) behave as if they are real orbits after reaching the chaotic sea. Here, we first discuss the applicability of this picture, particularly to the case with sharply divided phase space, and then move to a topic of dynamical tunneling in many dimensional systems, for which potentially existing complex orbits come out when the localization-delocalization transition occurs in the chaotic region. |
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