Super-fast spreading of billiard orbits in rational polygons and geodesics on translation surfaces

TL;DR

This paper demonstrates that billiard orbits in rational polygons and geodesics on translation surfaces spread rapidly, showing a form of uniform distribution in almost all directions, which advances understanding of their dynamical behavior.

Contribution

It introduces a super-fast spreading property for billiard orbits and geodesics, providing a new quantitative understanding of their uniform distribution in rational polygons and translation surfaces.

Findings

Billiard orbits exhibit super-fast spreading in rational polygons.

Geodesics on translation surfaces show rapid distribution in almost all directions.

The results imply optimal time-quantitative majority properties.

Abstract

In this paper, we show that billiard orbits in rational polygons and geodesics on translation surfaces exhibit super-fast spreading, an optimal time-quantitative majority property about the corresponding linear flow that implies uniformity in almost every direction.