Limiting distributions of curves under geodesic flow on hyperbolic manifolds

TL;DR

This paper studies how segments of curves on hyperbolic manifolds distribute under geodesic flow, showing they tend to become uniformly spread out unless constrained by geometric boundaries, using advanced dynamical systems theory.

Contribution

It demonstrates the asymptotic equidistribution of curves under geodesic flow on hyperbolic manifolds and characterizes the geometric conditions affecting this distribution.

Findings

Curves not in stable leaves become equidistributed under geodesic flow.

Lifted curves map into proper subspheres of the ideal boundary if constrained.

Equidistribution occurs unless the curve is geometrically restricted.

Abstract

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold. Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1(H^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\partial H^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in $H^n$ covering the closed immersed submanifold. In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\partial H^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure. The proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of SO(n,1).