# Horocycle Flows on Surfaces with Infinite Genus

## 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.

## Notes

### Acknowledgements

This set of notes constituted the basis for a series of lectures given in April 2015 as part of the program “Geometric and ergodic aspects of group actions,” at the Tata Institute for Fundamental Research, Mumbai. The author would like to thank the organizers of the program and the staff of TIFR for the kind hospitality. The author acknowledges the support of ISF grants 1149/18 and 199/14.

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