Minimal Lyapunov exponents, quasiconformal structures, and rigidity of non-positively curved manifolds

Abstract

For a closed irreducible non-positively curved manifold M, we show that if at almost every point one of the positive Lyapunov exponents for the geodesic flow achieves the minimum allowed by the curvature, then M is locally symmetric of non-compact type. Among the applications of this result, we show that rank one symmetric spaces may be characterized among negatively curved Hadamard manifolds admitting a cocompact lattice solely by the quasiconformal structure and Hausdorff dimension of their ideal boundary. We also prove a rigidity result for semiconjugacies.