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Indiscriminate covers of infinite translation surfaces are innocent, not devious

  • W. PATRICK HOOPER (a1) (a2) and RODRIGO TREVIÑO (a3)

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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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