Quantitative behavior of non-integrable systems (IV)

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.
Comment: 120 pages, 124 figures