A note taken from Section 20.5, Introduction to the Morden Theory of Dynamical Systems (by Katok und Hasselblatts).

Let be a closed manifold and be a transitive Anosov flow, that is, there exists a continuous invariant splitting such that, gives the flow direction, vectors in get uniformly contracted and vectors in get uniformly expanded. Let be the strong stable foliation tangent to , be the strong unstable foliation tangent to , be the stable foliation tangent to , be the unstable foliation tangent to . It is well known all leaves in these foliations are uniformly submanifolds.

We denote by if is contained in a single leaf, say , of . Denote by if is defined on a single leaf, say , of .

All ordinary notations are extended to this leaf-version, like openness, compactness, continuity, measurability and integrability.

Denote by the set of leaf-functions with compact and on which is continuous.

Note that may not be a linear space, since may not lie in , where . Still a functional is said to be linear if

(a). for all ,

(b). for all with .

Let be the collection of such linear functionals.

Fix a leaf-open set and a with . Consider a parameter family given by and the convex .

Proposition.

1. For each , there exists such that for all .

2. If is nonnegative and not identically zero, then there exists such that for all .

3. For each and , there exists such that for all -close to and all .

Theorem. There exists and a constant such that .

We want to use Riesz Representation Theorem to pick a measure on corresponding to . But the underline space is not linear space. So we start with a local plaque. Let be the collection of open -plaques with compact closures. Pick one such plaque and consider be the collection of continuous functions supported on . As a continuous linear functional on , the action of can be extended to , the usual space of continuous functions on and hence a leaf-measure (now by Riesz Representation Theorem). After checking the compactibility of the family , we get a system of leaf-measures with the following properties:

1. The restriction of on gives rise to a single, -finite measure on .

2. for all and all .

3. If are -equivalent, then .

Further we induces a measure on by flowing forward some time. Similarly we get a measure on (by reversing the time) Denote the corresponding rate by .

Patching these together, we will get the so called **Margulis measure** on , with local product structure and . Testing we get that (from now on we denote the common value by ) and hence the invariance of : .

Again by the local product structure we see and hence the metric entropy (for example, by Brin-Katok local entropy formula). Moreover it follows that , the topological entropy of . So this Margulis measure coincides with Bowen measure (measure of maximal entropy).

The following part taken from Margulis’s paper [M] “Certain measures associated with U-flows on compact manifolds”.

Let’s define . In particular there exist and such that for every . Dually we have for every and every . Two sets are said to be -equivalent if the -holonomy between is a homeomorphism and .

Lemma 2.1 in [M]. Let be a leaf-open subset. Then there exist and , such that for every , the set is -equivalent to some subset of .

It is equivalent to say that is minimal. Note that the here may be large. The

Lemma 2.3 in [M]. For each there exists such that for every -equivalent .

Lemma 2.4 in [M]. Let be a leaf-open subset. For given by Lemma 2.1, there exists such that for every and , .

Let and define (if ). Then let be the set of points with . Let with . Clearly and hence . In particular

Lemma 2.6 in [M]. For every , .

Let , be leaf-open and be given as above. Define a map . Clearly is continuous and positive on .

Lemma 2.8 in [M]. For every and , .

Proof. Applying Fubini Theorem and Lemma 2.4, we get

.

Two functions are -equivalent if

(a). and are -equivalent (via stable holonomy ),

(b). .

Lemma 2.9 in [M]. For every nonnegative and nonzero , and every leaf-compact , there exists such that for every bounded, leaf-measurable vanishing outside , and every ,

.

Proof. Pick such that has nonmepty leaf-interior. Then . Then set .

Lemma 2.11 in [M]. For any there exists $\epsilon>0$ such that for all -equivalent , .