I have been learning a bit about stable and unstable manifold theory for a non-uniformly hyperbolic diffeomorphism $f: M \to M$ on a smooth manifold. It seems that there are two completely separate cases, each having its own universe of literature: the case where $f$ is $C^1$ and the case where $f$ is $C^{1 + \alpha}$. I can understand more or less why the cases are so different, but I don't understand why anybody cares about the $C^1$ case (or really anything below $C^\infty$). The important examples with which I am familiar are the geodesic flow on a compact Riemannian manifold and a symplectomorphism on a symplectic manifold. Riemannian geometry is garbage below $C^2$, and I have trouble believing symplectic geometry is any better. So why all the fuss?

Thanks in advance!