Quantitative behavior of nonintegrable systems (III)
Abstract
The main purpose of part (III) is to give explicit geodesics and billiard orbits in polysquares and polycubes that exhibit timequantitative density. In many instances of the 2dimensional case concerning finite polysquares and related systems, we can even establish a best possible form of timequantitative density called superdensity. In the more complicated 3dimensional 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 timequantitative 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 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.06213
 Bibcode:
 2020arXiv200606213B
 Keywords:

 Mathematics  Number Theory;
 11K38;
 37E35
 EPrint:
 92 pages, 71 figures