Unique ergodicity of translation surfaces under branched n-covers

Abstract. We present a new construction on translation surfaces called the branched slit-induced (n)-cover: on a uniquely ergodic (X), pick a slit (s=[P,Q]); take (n) copies and switch sheets (i \mapsto i+1 \pmod n) each time the vertical flow hits (s) (i.e., glue the copies together). It turns out that the unique-ergodicity property is robust for such covers under fairly weak constraints. Moreover, the conditions are geometric despite the measure-theoretic core of the problem. This is especially notable because the setup is starkly different from the standard one in this area, where the varied parameter is not the flow direction but the surface construction itself. Joint with E.~Shuvaeva.

Recommended citation: Polina Baron and Elizaveta Shuvaeva. (2025). "Unique ergodicity of translation surfaces under branched n-covers." In preparation.
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