Let $M$ be the double-torus with a hyperbolic Riemannian metric. The geodesic flow on the unit tangent bundle $T^1M$ has many invariant Borel probability measures. In particular there are closed geodesics projecting to each non-trivial homotopy class of $M$, and one can support an invariant probability measure on each of these. Also one can take convex linear combinations of these invariant measures.

My question is the following: Are these the only invariant measures of zero metric (Kolmogorov-Sinai) entropy?

More generally, what are the zero entropy invariant probability measures of an Anosov geodesic flow?

Also I'm interested in the same question for shifts on finitely many symbols (i.e. What are the zero entropy invariant measures?).

Besides references giving an answer, other related references are of course very welcome.