Historic behaviour vs. physical measures for irrational flows with multiple stopping points

TL;DR

This paper investigates the behavior of Birkhoff averages in irrational flows on a torus with stopping points, revealing how Diophantine properties influence historic behavior and convergence of averages.

Contribution

It characterizes the dependence of Birkhoff average limits on Diophantine properties and the positions of stopping points in irrational flows with multiple stopping points.

Findings

Birkhoff limits diverge almost everywhere for Diophantine slopes.

Convergence of averages occurs for certain Liouville slopes and point configurations.

The behavior depends intricately on Diophantine properties and point positions.

Abstract

We study Birkhoff averages along trajectories of smooth reparameterizations of irrational linear flows of the two torus with two stopping points, say $\mathbf p$ and $\mathbf q$, of quadratic order. The limiting behaviour of such averages is independent of the starting point in a set of full Haar-Lebesgue measure and depends in an intricate way on the Diophantine properties of both the slope $\alpha$ of the linear flow as well as the relative position of $\mathbf p$ and $\mathbf q$. In particular, if $\alpha$ is Diophantine, then Birkhoff limits diverge almost everywhere (historic behaviour) and if $\alpha$ is sufficiently Liouville, then there exists some $\mathbf p$ and $\mathbf q$ such that the Birkhoff averages converge almost everywhere (unique physical measure).