Quantum transport through cantori

We study the effect of classical cantori in quantum mechanics, extending previous results by several groups. We find that cantori form exponential barriers to quantum transport not only when Planck's constant exceeds the flux through the cantorus but also when it is smaller than the flux. The mechanism of localization in the two cases is different, and we describe the switch from dynamical localization to a mechanism we call "retunneling" as Planck's constant increases. We investigate the Planck's over 2pi dependence of the exponential decay for retunneling and find that the Planck's over 2pi(-0.66) coefficient found previously at criticality appears to hold also away from criticality provided piPlanck's over 2pi is large enough compared to the flux. Numerical evidence as well as an analytic argument are given. Our final contribution to this subject is a phase space view of cantori in quantum mechanics. We illustrate our results using the whisker map.