Indiscriminate covers of infinite translation surfaces are innocent, not devious

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 Teichmüller flow and ergodic properties of the straight-line flow.

Improved exposition thanks to referee's comments. 54 pages, 9 figures

References in corpus (5)

• On the Ergodicity of Flat Surfaces of Finite Area
• Ergodicity for Infinite Periodic Translation Surfaces
• Flat surface models of ergodic systems
• Immersions and the space of all translation structures
• Flat surfaces, Bratteli diagrams, and unique ergodicity à la Masur

Cited by in corpus (1)

• Flat surfaces, Bratteli diagrams, and unique ergodicity à la Masur