This is a note taken from V. Kaimanovich’s paper *Bowen-Margulis and Patterson measures on negatively curved compact manifolds*.

Let be a simply connected negatively curved manifold (the exponential map turns out to be a diffeomorphism for every by Cartan-Hadamard Theorem). Two geodesic rays on are called asymptotic if they they have a bounded distance in the future. Denote by the set of asymptotic classes of geodesic rays on . Then is called the visibility compactification of . Let .

Remark: Another view point is . Clearly it is independent of .

**Definition**. A family of finite measures on is said to be -conformal of dimension if

(1) all these measures are equivalent (hence a.e. is meaningful without referring a special one)

(2) for a.e. , and for all .

Note that is given by a cocycle formula. So for arbitrary finite measure on and any point , the family -conformal of dimension .

Now let be a compact manifold with as its universal cover. Let be the fundamental group of . Note that

(3). each preserves the geodesics on and hence induces a map on .

**Theorem 1**. There exists a natural, 1 to 1 corrrespondence between the -covariant -conformal families of dimension on and the -conformal families.

Fix a point . Denote by the infimum such that Poincare series converges for all . Clearly does not depends on the choices of .

Remark. It is easy to see .

Consdier the probability , where . Let be a weak limit point of as . Note that

(4). the measure is concentrated on since it assigns zero measure to the fundamental domain.

Check that . So the weak limit is -conformal and hence (by Theorem 1) determines a -invariant, -conformal family of dimension on .

**Proposition 2**. Assume that the sectional curvatures of is bounded from above by . Then there exists a natural convex isomorphism between the invariant measures of the geodesic flow on and the -invariant finite measures on : induced by the fibration .

**Proposition 3**. Every -invariant, -conformal family on induces a -invariant finite measure on , and hence a invariant measure of the geodesic flow on .

*Proof*. Let be a measure on . Then . In particular the measure is independent of and hence a uniquely defined measure.

**Theorem 4**. The weak limit of corresponds to the measure of maximal entropy of the geodesic flow on .

In particular the following limit exists, and is called Patterson measure.

*Proof*. It suffices to show that the measure corresponding to via Proposition 3 has the following Margulis property:

() the conditional measures of on strong stable (and strong unstable) foliations get contracted by a factor (respectively, ) under the time-t geodesic flow.

Pick a point . Define the -holonomy , sending a point to . Similarly , sending a point to . Then the conditional measures are given by and .

Note that if , then and . Then () follows from the -conforamlity of .

Remark. Although the Liouville measure is always preserved by the geodesic flow, it may not be the measure of maximal entropy. In this case, these two measures are singular with respect to each other. Moreover the conditional measures of m.m.e. is also singular with respect to the leaf-volume. So is the Patterson measure on .