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Limiting distributions of curves under geodesic flow on hyperbolic manifolds. (English) Zbl 1171.37003

This paper (and its sequel) are concerned with different equidistribution properties of curves under geodesic flow. Consider the geodesic flow on the unit tangent bundle to a hyperbolic manifold of finite volume and a compact analytic curve not contained in any stable leaf for the flow. The main result here is that the normalized parameter measure on the curve becomes asymptotically equidistributed with respect to the normalized Riemannian measure on the unit tangent bundle of some closed totally geodesically immersed submanifold. Further analysis in the case of the submanifold being a proper subset is made via a lift of the curve to the universal covering space. The approach uses the dynamics of unipotent flows on homogeneous spaces.

MSC:

37A17 Homogeneous flows
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
22E40 Discrete subgroups of Lie groups

References:

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