Characterizing symmetric spaces by their Lyapunov spectra

TL;DR

This paper demonstrates that the Lyapunov spectra of periodic orbits uniquely characterize negatively curved locally symmetric spaces up to isometry, introducing new invariants and extending rigidity theorems for geodesic flows.

Contribution

It introduces the horizontal measure and horizontal dimension as new invariants, linking Lyapunov spectra to geometric rigidity and extending curvature pinching results.

Findings

Lyapunov spectra characterize spaces up to isometry.

Introduction of the horizontal measure and horizontal dimension.

Rigidity theorem for orbit equivalent Anosov flows.

Abstract

We prove that closed negatively curved locally symmetric spaces are characterized up to isometry among all homotopy equivalent negatively curved manifolds by the Lyapunov spectra of the periodic orbits of their geodesic flows. This is done by constructing a new invariant measure for the geodesic flow that we refer to as the horizontal measure. We show that the Lyapunov spectrum of the horizontal measure alone suffices to locally characterize these locally symmetric spaces up to isometry. We associate to the horizontal measure a new invariant, the horizontal dimension. We tie this invariant to extensions of curvature pinching rigidity theorems for complex hyperbolic manifolds to pinching rigidity theorems for the Lyapunov spectrum. Our methods extend to give a rigidity theorem for smooth Anosov flows $f^{t}$ orbit equivalent to the geodesic flow $g^{t}_{X}$ of a closed negatively curved locally symmetric space $X$: $f^{t}$ is smoothly orbit equivalent to $g^{t}_{X}$ if and only if its Lyapunov spectra on all periodic orbits are a multiple of the corresponding Lyapunov spectra for $g^{t}_{X}$.