The asymptotic behavior of geodesic circles in 2-torus of revolution and a sub-ergodic property
Poincaré recurrence implies that on a complete Riemannian manifold \(M\) with finite volume, given points \(p,q\in M\), there are sequences of unit vectors \(x_n\in SM\) and times \(t_n\to+\infty\) such that the footpoints of \(x_n\) converge to \(p\) and \(\gamma_{x_n}(t_n)\to q\), where \(\gamma_x\) is the geodesic with initial condition \(\dot\gamma_x(0)=x\). This motivates the author to study similar recurrence properties for the geodesic flow of manifolds with finite volume, particularly tori. Such properties are related to the existence of poles (points at which the exponential map is surjective) and ergodicity of the geodesic flow.
scientific article; zbMATH DE number 6148891