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A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds. (English) Zbl 1358.37014

H. Furstenberg [Lect. Notes Math. 318, 95–115 (1973; Zbl 0256.58009)] showed that the Liouville measure is the unique invariant measure for the unstable horocycle flow on the unit tangent bundle of a compact hyperbolic surface, and this result has been extended to several noncompact settings. Among these results S. G. Dani [Invent. Math. 47, 101–138 (1978; Zbl 0368.28021)] showed that (after excluding measures supported on periodic orbits) the Liouville measure is the only finite invariant ergodic measure on finite volume hyperbolic surfaces, and M. Burger [Duke Math. J. 61, No. 3, 779–803 (1990; Zbl 0723.58041)] showed there is a unique locally finite ergodic invariant measure on convex co-compact hyperbolic surfaces. T. Roblin [Mém. Soc. Math. Fr., Nouv. Sér. 95, 96 p. (2003; Zbl 1056.37034)] showed a more general result concerning unique ergodic Radon invariant measures for unstable horocycle flows. Here a new and simpler proof of the result of Roblin is given, by showing that the horospherical group acting on the frame bundle \(\Gamma\backslash \mathrm{SO}_o(d,1)\) of a hyperbolic manifold with infinite volume admits a unique invariant ergodic measure up to a scalar multiple supported on the set of frames whose geodesic orbits return infinitely often to a compact set.

MSC:

37A17 Homogeneous flows
22E40 Discrete subgroups of Lie groups
22D40 Ergodic theory on groups
28D15 General groups of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
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