Indiscriminate covers of infinite translation surfaces are innocent, not devious

TL;DR

This paper investigates how passing to finite covers affects the ergodic properties of straight-line flows on infinite translation surfaces, revealing that while ergodicity can be lost, it often remains preserved in random covers.

Contribution

It introduces a natural notion of random finite covers and shows that ergodicity is typically preserved under these covers, contrasting with known examples where ergodicity is destroyed.

Findings

Passing to finite covers can destroy ergodicity.

In many cases, ergodicity is preserved under random covers.

Provides new insights into the relationship between Teichmüller flow recurrence and ergodic properties.

Abstract

We consider the interaction between passing to finite covers and ergodic properties of the straight-line flow on finite area translation surfaces with infinite topological type. Infinite type provides for a rich family of degree $d$ covers for any integer $d>1$. We give examples which demonstrate that passing to a finite cover can destroy ergodicity, but we also provide evidence that this phenomenon is rare. We define a natural notion of a random degree $d$ cover and show that, in many cases, ergodicity and unique ergodicity are preserved under passing to random covers. This work provides a new context for exploring the relationship between recurrence of the Teichm\"uller flow and ergodic properties of the straight-line flow.