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Indiscriminate covers of infinite translation surfaces are innocent, not devious

Published online by Cambridge University Press:  04 December 2017

W. PATRICK HOOPER
Affiliation:
The City College of New York, New York, NY 10031, USA CUNY Graduate Center, New York, NY 10016, USA email whooper@ccny.cuny.edu
RODRIGO TREVIÑO
Affiliation:
Department of Mathematics, University of Maryland, College Park, USA email rodrigo@math.umd.edu

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.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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