Non-hyperbolic behavior of geodesic flows of rank 1 surfaces

TL;DR

This paper investigates the dynamics of geodesic flows on rank 1 surfaces, showing that in certain expansive but non-Anosov cases, the set of vectors with zero Lyapunov exponents has large Hausdorff dimension.

Contribution

It demonstrates that for specific rank 1 surfaces, the set of zero Lyapunov exponent vectors has large Hausdorff dimension, revealing complex dynamical behavior.

Findings

Large Hausdorff dimension of zero Lyapunov exponent vectors

Expansive but non-Anosov geodesic flows exhibit non-hyperbolic behavior

Provides insight into the structure of geodesic flows on rank 1 surfaces

Abstract

We prove that for the geodesic flow of a rank 1 Riemannian surface which is expansive but not Anosov the Hausdorff dimension of the set of vectors with only zero Lyapunov exponents is large.