Quantitative behavior of nonintegrable systems (III)
Abstract
The main purpose of part (III) is to give explicit geodesics and billiard orbits in polysquares that exhibit timequantitative density. In many instances, we can even establish a best possible form of timequantitative density called superdensity. We also study infinite flat dynamical systems, both periodic and aperiodic, which include billiards in infinite polysquare regions. 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 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.
 Publication:

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

 Mathematics  Number Theory;
 11K38;
 37E35
 EPrint:
 54 pages, 55 figures