Time-quantitative density of non-integrable systems

TL;DR

This paper introduces a novel method to analyze the time-quantitative density of non-integrable systems, specifically focusing on geodesics on flat surfaces, with applications to algebraic polyrectangle surfaces.

Contribution

The paper provides a new approach to establish time-quantitative density in flat dynamical systems and extends previous results to algebraic polyrectangle surfaces.

Findings

Half-infinite geodesics with badly approximable slopes are superdense.

New proof simplifies understanding of geodesic density.

Method applicable to algebraic polyrectangle surfaces.

Abstract

We introduce a new method to establish time-quantitative density in flat dynamical systems. First we give a shorter and different proof of our earlier result that a half-infinite geodesic on an arbitrary finite polysquare surface P is superdense on P if the slope of the geodesic is a badly approximable number. We then adapt our method to study time-quantitative density of half-infinite geodesics on algebraic polyrectangle surfaces.