Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds - II

TL;DR

This paper investigates the long-term behavior of smooth curves under geodesic flow on hyperbolic manifolds, showing convergence of measures and analyzing shrinking segments using advanced ergodic theory techniques.

Contribution

It extends previous results to C^n-smooth curves, establishing measure convergence and analyzing shrinking segments with new geometric and dynamical insights.

Findings

Measure on curves converges to the Riemannian measure under flow

Limits of shrinking segments are characterized

Uses Ratner's classification and SL(2,R) dynamics

Abstract

Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain natural geometric conditions, 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 SO(n,1), and an observation relating local growth properties of smooth curves and dynamics of linear SL(2,R)-actions.