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arXiv · 1503.00389

Indiscriminate covers of infinite translation surfaces are innocent, not devious

W. Patrick Hooper·Rodrigo Treviño
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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.

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W. Patrick Hooper, Rodrigo Treviño. 2015-03-02. Indiscriminate covers of infinite translation surfaces are innocent, not devious. https://doi.org/10.1017/etds.2017.120

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