Approximate control of the marked length spectrum by short geodesics

TL;DR

This paper demonstrates that approximate measurements of the marked length spectrum on a large finite set can nearly determine the Riemannian metric of a negatively curved manifold, using dynamical methods and explicit geometric dependencies.

Contribution

It extends the known uniqueness results of the MLS by showing approximate data on finite sets suffices to recover the metric, with explicit dependence on geometric parameters.

Findings

Approximate MLS on finite sets determines the metric up to small errors.

Constants depend only on geometric bounds like curvature and injectivity radius.

Dynamical tools are used to relate MLS closeness to metric closeness.

Abstract

The marked length spectrum (MLS) of a closed negatively curved manifold $(M, g)$ is known to determine the metric $g$ under various circumstances. We show that in these cases, (approximate) values of the MLS on a sufficiently large finite set approximately determine the metric. Our approach is to recover the hypotheses of our main theorems in arXiv:2203.12128, namely multiplicative closeness of the MLS functions on the entire set of closed geodesics of $M$. We use mainly dynamical tools and arguments, but take great care to show the constants involved depend only on concrete geometric information about the given Riemannian metrics, such as the dimension, sectional curvature bounds, and injectivity radii.