TITLE: What are the zero entropy invariant measures for an Anosov geodesic flow?
QUESTION [13 upvotes]: 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.

REPLY [8 votes]: Another type of answer. It can be shown that generically (in the sense of Baire), an invariant probability measure for the geodesic flow is ergodic, of full support, with entropy zero, and not mixing. See the article: 
 K. Sigmund. On the space of invariant measures for hyperbolic flows, Amer. J.Math. 94 (1972), 31–37
For shifts, it is due to Oxtoby 
J. C. Oxtoby. On two theorems of Parthasarathy and Kakutani concerning the shift transformation, (1963) Ergodic Theory (Proc. Internat. Sympos., Tulane Univ., New Orleans, La., 1961) pp. 203–215 Academic Press, New York.
These results can be generalized to any nonelementary negatively curved manifold, or shifts with infinitely many symbols, see for example " Generic measures for hyperbolic flows on non compact spaces", written with Yves Coudene   ,  Israël Journal of Maths vol 179 (2010). (The proof that in this situation, generic measures have entropy zero is not written in this article, and still unpublished, but works without any difficulty.)