Gap distributions and homogeneous dynamics

TL;DR

This paper surveys how $SL(2, )$ dynamics explain gap distributions in sequences from dynamical systems, providing a unifying theorem for various examples.

Contribution

It introduces an abstract theorem that unifies the understanding of gap distributions via homogeneous dynamics.

Findings

Unified explanation for gap distributions in different systems

Theorem connecting dynamics with statistical properties of sequences

Insights into trajectories of 2D dynamical systems

Abstract

We survey the use of dynamics of $SL(2, \R)$-actions to understand gap distributions for various sequences of subsets of $[0, 1)$, particularly those arising from special trajectories of various two-dimensional dynamical systems. We state and prove an abstract theorem that gives a unified explanation for some of the examples we present.