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Horocycle flow on geometrically finite surfaces. (English) Zbl 0723.58041

Suppose \(S=\Gamma \setminus D^ 2\) is a quotient of the Poincaré disc by a finitely generated discrete group \(\Gamma\) of orientation preserving isometries acting without fixed points on \(D^ 2\). The group of orientation preserving isometries of \(D^ 2\) is \(PSL(2,{\mathbb{R}})\) and the unit tangent bundle \(T_ 1S\) of S is a homogeneous space of \(PSL(2,{\mathbb{R}}):\) \(T_ 1S=\Gamma \setminus PSL(2,{\mathbb{R}}).\) The unipotent subgroup of PSL(2,\({\mathbb{R}})\), \(N=\{n(x)=\begin{pmatrix} 1&x \\ 0&1 \end{pmatrix}:\) \(x\in {\mathbb{R}}\}\) acts on \(T_ 1S\). The author determines all N-invariant Radon measures on \(T_ 1S\).

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
58C35 Integration on manifolds; measures on manifolds
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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