Orbits of horospherical flows
A result of \textit{G. A. Hedlund} [ibid. 2, 530-542 (1936; Zbl 0015.10201)] asserts that every orbit of the horocycle flow associated to a surface of constant negative curvature and finite area is either dense or periodic. If the surface is compact then every orbit is dense while if the surface is noncompact then there are as many one-parameter families of periodic orbits as the number of cusps. The horocycle flow can be regarded as the action of the one-parameter subgroup \(U=\{\left( \begin{matrix} 1\quad t\\ 0\quad 1\end{matrix} \right)|\) \(t\in {\mathbb{R}}\}\) of \(G=SL(2,{\mathbb{R}})\) on the homogeneous space G/\(\Gamma\) where \(\Gamma\) is an appropriate lattice in G. In this paper a generalization of Hedlund's theorem is obtained. The main result asserts that if G is a connected reductive Lie group, \(\Gamma\) a lattice in G, U a horospherical subgroup of G, and \(x\in G/\Gamma\) then there exists a closed subgroup H of G containing U such that the closure of Ux is Hx and such that Hx admits a finite H-invariant measure.
scientific article; zbMATH DE number 3986242
scientific article