Invariant measures for the horocycle flow on periodic hyperbolic surfaces |
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Authors: | François Ledrappier Omri Sarig |
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Institution: | (1) Department of Mathematics, University of Notre-Dame, Notre-Dame, IN 46556-4618, USA;(2) Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA |
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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.
The first author was supported by NSF grant 0500630.
The second author was supported by NSF grant 0400687. |
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