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.



What are the zero entropy invariant measures for an Anosov geodesic flow?
There are lots of zero entropy invariant probability measures, many more than just the obvious ones supported on periodic orbits. As you suggest in the question, one can understand the general case by just considering what happens for symbolic systems.


**Explicit example:** Let $\alpha$ be irrational and let $a\_n$ denote the fractional part of $n\alpha$. Consider the sequence in $\Sigma\_2 = \{0,1\}^\mathbb{Z}$ given by $x\_n = 0$ if $0 \leq a\_n < 1/2$, and $x\_n=1$ if $1/2\leq a\_n<1$. Let $X\subset \Sigma\_2$ be the orbit closure of $x=(x\_n)$; then there is an entropy-preserving isomorphism between the space of invariant measures for the shift map $\sigma\colon X\to X$ and for the irrational rotation $R\_\alpha\colon S^1 \to S^1$. The latter preserves Lebesgue measure on the circle and is uniquely ergodic with zero entropy, so $X$ supports exactly one invariant probability measure $\mu$, which comes from Lebesgue and has zero entropy. Now $\mu$ is a shift-invariant probability measure on $\Sigma\_2$ that has zero entropy but is not supported on a periodic orbit.


**General result:** In fact, the above construction is representative of a general phenomenon. As RW points out in his answer, you can get lots of zero entropy measures on shift spaces by taking generating partitions for zero entropy transformations. You can even get more, using the Jewett-Krieger embedding theorem (see Petersen's "Ergodic Theory" or Denker, Grillenberger, and Sigmund's "Ergodic Theory on Compact Spaces"), which lets you find a closed shift-invariant subset of the shift space that has the desired measure as its only shift-invariant probability measure. So there's a lot there.


