Quantitative behavior of non-integrable systems (III).

The main purpose of the paper is to give explicit geodesics and billiard orbits in polysquares and polycubes that exhibit time-quantitative density. In many instances of the 2-dimensional case concerning finite polysquares and related systems, we can even establish a best possible form of time-quantitative density called superdensity. In the more complicated 3-dimensional case concerning finite polycubes and related systems, we get very close to this best possible form, missing only by an arbitrarily small margin. We also study infinite flat dynamical systems, both periodic and aperiodic, which include billiards in infinite polysquares and polycubes. In particular, we can prove time-quantitative density even for aperiodic systems.