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Invariant measures for the horocycle flow on periodic hyperbolic surfaces

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Abstract

We classify the ergodic invariant Radon measures for the horocycle flow on geometrically infinite regular covers of compact hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the surface. Two consequences arise: if the group of deck transformations G is of polynomial growth, then these measures are classified by the homomorphisms from G 0 to ℝ where G 0 ≤ G is a nilpotent subgroup of finite index; if the group is of exponential growth, then there may be more than one Radon measure which is invariant under the geodesic flow and the horocycle flow. We also treat regular covers of finite volume surfaces.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre-Dame, Notre-Dame, IN, 46556-4618, USA

    François Ledrappier

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

    Omri Sarig

Authors
  1. François Ledrappier
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  2. Omri Sarig
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Additional information

The first author was supported by NSF grant 0500630.

The second author was supported by NSF grant 0400687.

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Ledrappier, F., Sarig, O. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Isr. J. Math. 160, 281–315 (2007). https://doi.org/10.1007/s11856-007-0064-0

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  • DOI: https://doi.org/10.1007/s11856-007-0064-0

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