Universal hitting time statistics for integrable flows

TL;DR

This paper demonstrates that integrable flows, despite lacking mixing properties, exhibit universal limit laws for hitting times, expanding understanding of probabilistic behavior in dynamical systems.

Contribution

It introduces new universal limit laws for hitting times in integrable flows, using a novel equidistribution theorem based on Ratner's measure classification.

Findings

Hitting times in integrable flows follow universal, non-Poisson limit laws.

The results apply to generic integrable flows and specific target sets.

A new equidistribution theorem in lattice space underpins the analysis.

Abstract

The perceived randomness in the time evolution of "chaotic" dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for "generic" integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner's measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.