Quantitative behavior of non-integrable systems (III)

TL;DR

This paper investigates the detailed behavior of billiard trajectories and geodesics in polysquare regions, demonstrating explicit density and superdensity properties, including for infinite and aperiodic systems, with implications for optical illumination.

Contribution

It provides explicit constructions of geodesics and billiard orbits exhibiting time-quantitative density and superdensity in both finite and infinite polysquare systems, including aperiodic cases.

Findings

Explicit geodesics and billiard orbits with density properties

Superdensity established for certain systems

Single initial directions can illuminate uncountably many systems

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.