## Some distinguished meausres

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

Let $M$ be a simply connected negatively curved manifold (the exponential map $\exp_x:T_xM\to M$ turns out to be a diffeomorphism for every $x$ by Cartan-Hadamard Theorem). Two geodesic rays on $M$ are called asymptotic if they they have a bounded distance in the future. Denote by $\partial M$ the set of asymptotic classes of geodesic rays on $M$. Then $\overline{M}=M\sqcup\partial M$ is called the visibility compactification of $M$. Let $\partial^2 M=\partial M\times \partial^2 M\backslash \triangle$.

• For each pair $(x,\alpha)\in M\times \partial M$, there exists a unique geodesic ray $\gamma=\gamma_{x,\alpha}\in\alpha$ with $\gamma(0)=x$.
• For each pair $(\alpha,\beta)\in \partial^2 M$, there exists a unique complete geodesic $\gamma$ with $\gamma(R_+)\in\alpha$ and $\gamma(R_-)\in\beta$.
• Each unit vector $\xi\in SM$ corresponds to a complete geodesic $\gamma_\xi$, and hence two maps $\xi\mapsto \alpha(\gamma_\xi)$ and $\xi\mapsto \beta(\gamma_\xi)$. This induces a $\mathbb{R}$-fibration $SM\to \partial^2 M$, $\xi\mapsto (\alpha(\gamma_\xi),\alpha(\gamma_\xi))$.
• For each pair $(x,\alpha)$, we pick the geodesic ray $\gamma$ and define Busemann function $b_{\alpha,x}:y\in M\mapsto \lim_{t\to\infty}(d(x,\gamma(t))-t)$. The level sets of $b_{\alpha,x}$ are the horospheres on $M$ centered at $\alpha\in\partial M$.
• More generally, let $b_\alpha(y,z)=b_{\alpha,x}(y)-b_{\alpha,x}(z)$ (need to check it is independent of the choice of $x$ and $\gamma_{\alpha,x}$) (Remark). Moreover, $b_\alpha(y,z)$ gives the signed distance between the horospheres passing through $y$ and $z$ centered at $\alpha$.
Remark: Another view point is $b_\alpha(y,z)=b_{\alpha,z}(y)$. Clearly it is independent of $x$.
• Define another function $B_x:(\alpha,\beta)\in\partial ^2M \mapsto b_\alpha(x,y)+b_\beta(x,y)$, where $y$ lies on the geodesic $\gamma_{\alpha,\beta}$. (Check it is independent of the choices of $y$ on that geodesic). Geometrically, it measures the length of the segment cut out from $\gamma$ by the horospheres passing through $x$ and centered at $\alpha$ and $\beta$. Moreover we have $B_x(\alpha,\beta)-B_y(\alpha,\beta)=b_{\alpha}(x,y)+b_{\beta}(x,y)$.
• Definition. A family $\{\mu_x:x\in M\}$ of finite measures on $\partial M$ is said to be $b$-conformal of dimension $h$ if
(1) all these measures are equivalent (hence a.e. is meaningful without referring a special one)
(2) $\frac{d\mu_x}{d\mu_x}(\alpha)=e^{h\cdot b_\gamma(x,y)}$ for a.e. $\alpha\in\partial M$, and for all $x,y\in M$.

Note that $b_\gamma$ is given by a cocycle formula. So for arbitrary finite measure $\mu$ on $\partial M$ and any point $p\in M$, the family $\mu_x=e^{h\cdot b_\gamma(x,p)}\mu$ $b$-conformal of dimension $h$.

Now let $N$ be a compact manifold with $M$ as its universal cover. Let $G$ be the fundamental group of $N$. Note that
(3). each $g\in G$ preserves the geodesics on $M$ and hence induces a map on $\partial M$.
Theorem 1. There exists a natural, 1 to 1 corrrespondence between the $G$-covariant $b$-conformal families of dimension $h$ on $\partial M$ and the $(G,b)$-conformal families.

Fix a point $p\in M$. Denote by $h=h(N)$ the infimum such that Poincare series $P(p,s,N)=\sum_{g\in G}e^{-s\cdot d(p,gp)}$ converges for all $s>h$. Clearly $h$ does not depends on the choices of $p\in M$.
Remark. It is easy to see $h=\lim_{R\to\infty}\frac{\log \text{vol}B(p,R)}{R}$.

Consdier the probability $\mu^s=\frac{1}{P(p,s,N)}\sum_{g\in G}e^{-s\cdot d(p,gp)}\delta_{gp}$, where $s>h$. Let $\mu$ be a weak limit point of $\mu^s$ as $s\to h$. Note that
(4). the measure $\mu$ is concentrated on $\partial M$ since it assigns zero measure to the fundamental domain.

Check that $\frac{d(g\mu)}{d\mu_x}(\alpha)=\lim_{y\to \alpha e^{h\cdot d(x,y)-h\cdot d(gx,y)}}e^{h\cdot b_\gamma(x,gx)}$. So the weak limit $\mu$ is $(G,b)$-conformal and hence (by Theorem 1) determines a $G$-invariant, $b$-conformal family of dimension $h$ on $\partial M$.

Proposition 2. Assume that the sectional curvatures of $N$ is bounded from above by $-1$. Then there exists a natural convex isomorphism between the invariant measures $d\lambda$ of the geodesic flow on $SN$ and the $G$-invariant finite measures $d\Lambda$ on $\partial^2M$: $d\lambda= dt\cdot d\Lambda$ induced by the fibration $SM\to \partial^2M$.

Proposition 3. Every $G$-invariant, $b$-conformal family on $\partial M$ induces a $G$-invariant finite measure on $\partial^2M$, and hence a invariant measure of the geodesic flow on $SN$.
Proof. Let $\Gamma_x=\mu_x\times \mu_x$ be a measure on $\partial^2M$. Then $\frac{d\Gamma_x}{d\Gamma_y}(\alpha,\beta)=\frac{d\mu_x}{d\mu_y}(\alpha)\cdot \frac{d\mu_x}{d\mu_y}(\beta)$ $=e^{h b_{\alpha}(x,y)+ h b_{\beta}(x,y)}=e^{h\cdot (B_x(\alpha,\beta)-B_y(\alpha,\beta))}$. In particular the measure $\Lambda=e^{h\cdot B_x} \Gamma_x$ is independent of $x$ and hence a uniquely defined measure.

Theorem 4. The weak limit $\mu$ of $\mu^s$ corresponds to the measure of maximal entropy of the geodesic flow on $SN$.
In particular the following limit $\mu=\lim_{s\to h}\mu^s$ exists, and is called Patterson measure.
Proof. It suffices to show that the measure $\nu$ corresponding to $\mu$ via Proposition 3 has the following Margulis property:
($\ast$) the conditional measures of $\nu$ on strong stable (and strong unstable) foliations get contracted by a factor $e^{-h t}$ (respectively, $e^{-h t}$) under the time-t geodesic flow.

Pick a point $\xi\in S_xN$. Define the $\infty$-holonomy $h^s_x:W^s(\xi)\to \partial M$, sending a point $\eta\in W^s(\xi)$ to $\gamma(\eta,-\infty)$. Similarly $h^u_x:W^u(\xi)\to \partial M$, sending a point $\eta\in W^u(\xi)$ to $\gamma(\eta,\infty)$. Then the conditional measures are given by $\nu_{W^s(\xi)}=h^s_x(\mu)$ and $\nu_{W^u(\xi)}=h^s_u(\mu)$.
Note that if $\eta=\phi_t(\xi)\in S_yN$, then $b_{\alpha(\xi)}(x,y)=-t$ and $b_{\beta(\xi)}(x,y)=t$. Then ($\ast$) follows from the $b$-conforamlity of $\mu$.

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 $\partial M$.