Abstract

We study the ergodic theory of horocycle flows on hyperbolic surfaces with infinite genus. In this case, the nontrivial ergodic invariant Radon measures are all infinite. We explain the relation between these measures and the positive eigenfunctions of the Laplacian on the surface. In the special case of \(\mathbb Z^d\)-covers of compact hyperbolic surfaces, we also describe some of their ergodic properties, paying special attention to equidistribution and to generalized laws of large numbers.

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