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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.

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Added ergodic theory tag. – Ian Morris Oct 20 '10 at 9:37
up vote 6 down vote accepted

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.

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Thank you! This answer is very clearly explained and convincing. I still feel it would be nice if there was some explicit reference or construction explaining how to pass these measures from the shifts over to a geodesic flow, however I've decided to accept this. – Pablo Lessa Oct 15 '10 at 21:48
@Pablo, it's a general theorem (the buzz-word here is Adler-Weiss, see for example this survey by Adler -… ) which tells you you can "encode" such systems and achieve a measurable isomorphism with such Bernoulli systems. From there, it is not hard to construct zero entropy measures as averaging over "thin" orbits (obviously those are shifts of finite type, so you need to do some massaging to get those "thin" sets then the full shift, but it's possible). – Asaf Jan 19 '12 at 18:28

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.)

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I just saw this... I have recently finished a paper with Terry Soo in which we prove a universality property for geodesic flows: for any measure preserving transformation $T\colon X\to X$ with (measure-theoretic) entropy less than the topological entropy of the time one map of the geodesic flow, there's an injective map from $X$ into the geodesic flow equipped with the time 1 map.

This gives a very rich family of invariant measures for the time 1 map of the geodesic flow. Averaging these measures over times from 0 to 1 gives a large collection of invariant measures for the full flow. In particular if you start off with a zero entropy transformation, the measure you obtain this way is a zero entropy measure for the flow.

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By taking a generating partition you can realize an arbitrary finite entropy transformation as the full shift on a finite alphabet with a certain invariant measure. In particular, this observation provides non-trivial zero entropy invariant measures for the full shift. This can also be done for subshifts of finite type, and therefore (in view of the symbolic business) for Anosov diffeomorphisms - there is an old paper by Lind and Thouvenot where they consider a similar realization problem for arbitrary finite entropy transformations.

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OK, I am sure that there must be something easier for the shifts but... In my paper with Glendinning we briefly mention a construction of a subshift on two symbols of zero topological entropy. Namely, let $\sigma$ denote the shifted Thue-Morse sequence, i.e., $$ \sigma = 1101001100101101001011001101001 \dots $$ Define $X$ to be the set of all two-sided 0-1 sequences $(x_n)$ such that $$ (x_n, x_{n+1},\dots) \prec \sigma \ \ \ \forall n\in\mathbb Z $$ and $$ (1-x_n, 1-x_{n+1},\dots) \prec \sigma \ \ \ \forall n\in\mathbb Z, $$ where $\prec$ stands for ``lexicographically smaller''. The set $X$ has zero Hausdorff dimension in the standard metric on $\{0,1\}^{\mathbb Z}$. In fact, the number of admissible words of length $n$ in $X$ grows approximately as $n^{\log n}$.

To endow $X$ with a shift-invariant measure, one can take the sequence of subshifts of finite type converging to $X$ (by truncating $\sigma$) and $\mu$ to be the weak limit of the corresponding measures of maximal entropy, say.

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