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The statistics of Poincar\'e recurrence times in Hamiltonian systemstypically shows a power-law decay with chaotic trajectories sticking to somephase-space regions for long times. For higher-dimensional systems themechanism of this power-law trapping is still unknown. We investigate trappedorbits of a generic 4D symplectic map in phase space and frequency space andfind that, in contrast to 2D maps, the trapping is (i) not due to a hierarchyin phase space. Instead, it occurs at the surface of the regular region, (ii)outside of the Arnold web. The chaotic dynamics in this sticky region is (iii)dominated by resonance channels which reach far into the chaotic region: Weobserve (iii.a) clear signatures of some kind of partial transport barriers andconjecture (iii.b) a stochastic process with an effective drift along resonancechannels. These two processes lay the basis for a future understanding of themechanism of power-law trapping in higher-dimensional systems.

What is the mechanism of power-law distributed Poincar\'e recurrences in higher-dimensional systems?