Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds

Abstract

Extending earlier results for analytic curve segments, in this article we describe the asymptotic behavior of evolution of a finite segment of a $C^n$-smooth curve under the geodesic flow on the unit tangent bundle of a hyperbolic $n$-manifold of finite volume. In particular, we show that if the curve satisfies certain natural geometric conditions, then the pushforward of the parameter measure on the curve under the geodesic flow converges to the normalized canonical Riemannian measure on the tangent bundle in the limit. We also study the limits of geodesic evolution of shrinking segments.

We use Ratner's classification of ergodic invariant measures for unipotent flows on homogeneous spaces of $\mathrm{SO}(n,1)$ and an observation relating local growth properties of smooth curves and dynamics of linear $\mathrm{SL}(2,\mathbb{R})$-actions