I have a question concerning the ergodic properties of hyperbolic Hamiltonian flows.
Let $\Phi_{H}^{t}$ be a Hamiltonian flow on a symplectic manifold $\mathcal{M}$. If $\Phi_{H}^{t}$ is Anosov on a compact and connected energy shell $\Sigma_{E}:=\{x\in\mathcal{M}:H(x)=E\}$, is it necessarily ergodic with respect to Liouville measure on $\Sigma_{E}$?
If not, does anyone know of any simple counterexamples? Also, which supplementary assumptions would one need in order to guarantee ergodicity?
It was proved by Anosov that any $C^2$ transitive Anosov flow (on a compact manifold) is ergodic (using the Hopf argument). The same can be done for $C^{1+\alpha}$ transitive Anosov flows, due to the fact that the stable and unstable foliations remain absolutely continuous.