The central limit theorem for the geodesic flow on noncompact manifolds of constant negative curvature
Let \(S\) be a manifold of finite volume and constant curvature \(-1\) with unit tangent bundle \(T^ 1 S\). The author derives a central limit theorem for the process on \(T^ 1 S\) which is obtained by integrating along the orbits of the geodesic flow \(\vartheta^ t\), a smooth function which together with its derivatives of all orders is square integrable with respect to the normalized Lebesgue-Liouville measure \(\lambda\). More precisely, he shows that for every such function \(\phi\) there is a number \(\sigma_ \phi \geq 0\) with the following properties: 1) \(\sigma_ \phi = 0\) only if \(\phi = X(\psi) + \text{const}\) where \(X\) is the geodesic spray and \(\psi\) a sufficiently regular function. 2) \(\lim_{t \to \infty} \int \exp (it^{-1/2} (\int^ t_ 0 \phi (\vartheta^ s v) ds - \int \phi d \lambda)) d\lambda(v) = (e^{- \sigma^ 2_ \phi /2})\). If \(\sigma_ \phi > 0\) then this means that the distribution of the variables \({1\over \sigma_ \phi \sqrt {t}} [\int^ t_ 0 \phi (\vartheta^ s v)ds - t \int \phi d\lambda]\) is asymptotically close to the normal distribution when \(t\) goes to infinity. The proof is carried out for hyperbolic surfaces \(S\). Let \(\omega\) be the canonical contact form on \(T^ 1 S\); there is a unique section \(\zeta_ \phi\) of the dual \(T^* W^ s\) of the tangent bundle of the stable foliation \(W^ s\) on \(T^ 1 S\) whose restriction to a stable manifold is closed. The stable Laplacian \(\Delta^ s\) on \(T^ 1 S\) induces a leafwise diffusion process with stationary measure \(\lambda\). A typical path of this diffusion follows an orbit of the geodesic flow, and this enables to show that \[ \lim_{t \to \infty} \int \exp \biggl( it^{- 1/2} \int^ t_ 0 \phi (\vartheta^ s v) ds\biggr) d\lambda(v) = \lim_{t \to \infty} \mathbb{E} (\exp (-it^{-1/2} \vartheta_ \phi (\xi_ t))) \] where, for a path \(\xi_ t\) on a stable manifold \(W^ s (\xi_ 0)\), one defines \(\vartheta_ \phi (\xi_ t) = \int^ t_ 0 \zeta_ \phi (\alpha' (s))ds\) where \(\alpha\) is a smooth path on \(W^ s (\xi_ 0)\) joining \(\alpha(0) = \xi_ 0\) to \(\alpha(1) = \xi_ t\) and homotopic to \(\xi|_{[0,t]}\) with fixed endpoints and \(\mathbb{E}\) denotes expectation. The theorem then follows by computing this expectation using the fact that the diffusion induced by \(\Delta^ s\) is exponentially contracting with respect to a suitable norm on the space of smooth square integrable functions on \(T^ 1 S\).
scientific article; zbMATH DE number 624875