2020/06/11 by J. Beck, W. W. L. Chen, Beck, J. +3
Materials Science · Mathematics · Physics and Astronomy · #11K38 #37E35 #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2006.06213
openalex publication_date 2020/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.