Quantitative behavior of non-integrable systems (III)
Abstract
The main purpose of part (III) is to give explicit geodesics and billiard orbits in polysquares that exhibit time-quantitative density. In many instances, we can even establish a best possible form of time-quantitative 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 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 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 e-prints
- Pub Date:
- June 2020
- arXiv:
- arXiv:2006.06213
- Bibcode:
- 2020arXiv200606213B
- Keywords:
-
- Mathematics - Number Theory;
- 11K38;
- 37E35
- E-Print:
- 54 pages, 55 figures