A geometric criterion on embedded disks along Teichmuller geodesics implies that for uniquely ergodic translation surfaces, almost every branched N-cover via a slit is uniquely ergodic.
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Hodge generic points of S over the algebraic numbers are analytically dense in S over the complex numbers for quasi-projective families of varieties.
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Unique ergodicity of branched covers of translation surfaces
A geometric criterion on embedded disks along Teichmuller geodesics implies that for uniquely ergodic translation surfaces, almost every branched N-cover via a slit is uniquely ergodic.