Quantitative behavior of non-integrable systems. III

The main purpose of part III 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 polysquare and polycube regions. In particular, we can prove time-quantitative density even for aperiodic systems. In terms of optics the billiard case is equivalent to the result that an explicit single ray of light can essentially illuminate a whole infinite polysquare or polycube region with reflecting boundary acting as mirrors. In fact, we show that the same initial direction can work for an uncountable family of such infinite systems. Some of these infinite systems belong to the class of Ehrenfest wind-tree models, introduced by physicists about 100 years ago. Thus we obtain, for the first time, explicit density results about the time evolution of these infinite aperiodic billiard models in physics. What makes our positive density results in the case of the periodic Ehrenfest wind-tree models particularly interesting is the recent discovery by Fraczek and Ulcigrai [8] about these models that for almost every initial direction, the billiard orbit is not dense. To prove density of explicit orbits, we use a non-ergodic method, which is an eigenvalue-free version of the shortline method. The original eigenvalue-based version of the shortline method, introduced and developed in [2,3], enables us to prove time-quantitative equidistribution of orbits. The reader does not need to be familiar with those long papers. Here we make a serious effort to keep this paper self-contained.