Quantitative behavior of non-integrable systems (IV)

TL;DR

This paper advances the analysis of non-integrable systems by simplifying and extending the surplus shortline method to a broader class of 2D flat dynamical systems, enabling more efficient determination of orbit irregularities and escape rates.

Contribution

It simplifies the eigenvalue-based surplus shortline method for finite polysquare translation surfaces and extends it to a wider class of 2D flat dynamical systems including Veech surfaces.

Findings

Simplified the surplus shortline method for finite polysquare translation surfaces.

Extended the method to include all Veech surfaces and other 2D flat systems.

Established quantitative results on orbit irregularity and escape rates.

Abstract

In this paper, there are two sections. In Section 7, we simplify the eigenvalue-based surplus shortline method for arbitrary finite polysquare translation surfaces. This makes it substantially simpler to determine the irregularity exponents of some infinite orbits, and quicker to find the escape rate to infinity of some orbits in some infinite models. In Section 8, our primary goal is to extend the surplus shortline method, both this eigenvalue-based version as well as the eigenvalue-free version, for application to a large class of 2-dimensional flat dynamical systems beyond polysquares, including all Veech surfaces, and establish time-quantitative equidistribution and time-quantitative superdensity of some infinite orbits in these new systems.