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The Horocycle Flow and the Laplacian on Hyperbolic Surfaces of Infinite Genus

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Abstract

Consider a complete hyperbolic surface which can be partitioned into countably many pairs of pants whose boundary components have lengths less than some constant. We show that any infinite ergodic invariant Radon measure for the horocycle flow is either supported on a single horocycle associated with a cusp, or corresponds canonically to an extremal positive eigenfunction of the Laplace–Beltrami operator.

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Authors and Affiliations

  1. Mathematics Department, Pennsylvania State University, University Park, PA, 16802, USA

    Omri Sarig

  2. The Weizmann Institute of Science, Rehovot, 76100, POB 26, Israel

    Omri Sarig

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  1. Omri Sarig
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Correspondence to Omri Sarig.

Additional information

The author was partially supported by NSF grant 0652966 and a Sloan Research Fellowship.

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Sarig, O. The Horocycle Flow and the Laplacian on Hyperbolic Surfaces of Infinite Genus. Geom. Funct. Anal. 19, 1757–1812 (2010). https://doi.org/10.1007/s00039-010-0048-9

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  • DOI: https://doi.org/10.1007/s00039-010-0048-9

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