A universal divergence rate for symmetric Birkhoff Sums in infinite ergodic theory

TL;DR

This paper establishes a universal gap in the failure of the ergodic theorem for symmetric Birkhoff sums in infinite ergodic theory and applies it to fluctuations of Birkhoff integrals in horocyclic flows.

Contribution

It introduces a universal divergence rate for symmetric Birkhoff sums and applies this to analyze fluctuations in horocyclic flow integrals.

Findings

Existence of a universal gap in symmetric Birkhoff sum failure

Application to fluctuations in horocyclic flows

Insights into ergodic theorem deviations in infinite systems

Abstract

We show that there exists a universal gap in the failure of the ergodic theorem for symmetric Birkhoff sums in infinite ergodic theory. In addition, an application of this result to a question of fluctuations of the Birkhoff integrals of horocyclic flows on geometrically finite surfaces is given.