Can Jacobi Fields be used to explicitly calculate Lyapunov exponents for geodesic flows?

Question

My background is in geometry, and I have just become interested in the ergodic theory of geodesic flow. In Sarnak's 1981 paper "Entropy estimates for geodesic flows," the following definition is given for the Lyapunov exponent of the geodesic flow.

Let $v \in SM$ (unit tangent bundle) and $\xi \in T_v (SM)$. The Lyapunov exponent at $v$ in the direction of $\xi$ is then $$\chi^+ (v,\xi) = \lim_{t\to \infty} \frac{1}{t} \log || d \phi_t (\xi)||$$ Where $\phi_t$ is the geodesic flow and $||\cdot||$ is the norm on $T(TM)$ induced via the connection.

There is an isomorphism between $T_v (TM)$ and the Jacobi fields along $\gamma$ (the geodesic with tangent vector $v$) set up by simply solving the Jacobi equation, with the added property that the norm of the pushfoward $d \phi_t$ along the flow for a certain $\xi$ can be related to the norms of the associated Jacobi field and its covariant derivative.

Does this mean that if the Jacobi equation can be solved, or inequlities developed from it, we can explicitly find Lyapunov exponents for the flow or at least estimate them? Also, how does integrability of the flow help in this direction?

For example, can we get the (positive) Lyapunov exponent for the geodesic flow on a compact surface of negative curvature by the above argument?

Answer 1

The answer is yes, although to give the details would require introducing too much material for an answer.

My suggestion is that you have a look at these references (in this order):

• Section 17.6 of the Katok-Hasselblatt book for negative sectional curvature;

• Section 12.1 of the Barreira-Pesin book for manifolds without conjugate points.

Both use delicate bounds to estimate the Lyapunov exponents, although the first reference is a bit less cryptic (it basically obtains estimates inside invariant cones, which then yield estimates for the Lyapunov exponents).

On the other hand, the second reference is much more general and gives rise to nonuniformly hyperbolic dynamics that need not be uniformly hyperbolic (although you only ask about the Lyapunov exponents).