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The dynamics in three-dimensional billiards leads, using a Poincar\'esection, to a four-dimensional map which is challenging to visualize. By meansof the recently introduced 3D phase-space slices an intuitive representation ofthe organization of the mixed phase space with regular and chaotic dynamics isobtained. Of particular interest for applications are constraints to classicaltransport between different regions of phase space which manifest in thestatistics of Poincar\'e recurrence times. For a 3D paraboloid billiard weobserve a slow power-law decay caused by long-trapped trajectories which weanalyze in phase space and in frequency space. Consistent with previous resultsfor 4D maps we find that: (i) Trapping takes place close to regular structuresoutside the Arnold web. (ii) Trapping is not due to a generalizedisland-around-island hierarchy. (iii) The dynamics of sticky orbits is governedby resonance channels which extend far into the chaotic sea. We find clearsignatures of partial transport barriers. Moreover, we visualize the geometryof stochastic layers in resonance channels explored by sticky orbits.

3D billiards: visualization of regular structures and trapping of chaotic trajectories