Superdensity and super-micro-uniformity in non-integrable flat systems

TL;DR

This paper demonstrates that in non-integrable flat systems, superdensity ensures an optimal level of uniformity called super-micro-uniformity, advancing understanding of their dynamical properties.

Contribution

It introduces the concept of super-micro-uniformity and establishes its connection to superdensity in non-integrable flat systems.

Findings

Superdensity implies super-micro-uniformity in non-integrable flat systems.

Super-micro-uniformity is an optimal form of time-quantitative uniformity.

The results apply to finite polysquare translation surfaces.

Abstract

We show that on any non-integrable finite polysquare translation surface, superdensity, an optimal form of time-quantitative density, leads to an optimal form of time-quantitative uniformity that we call super-micro-uniformity.