Tema 12: Busemann densities

Given a Hadamard manifold $X$ and a discrete group of isometries $\Gamma$, a family of Borel measures $\{\mu_p\colon p\in X\}$ on $X(\infty)$ is an $\alpha$-dimensional Busemann density if it satisfies:

• ${\rm supp}\mu\subset\Lambda(\Gamma)$,
• $\frac{d\mu_p}{d\mu_q}(\xi)=e^{-\alpha b_p(q,\xi)}$ for almost every $\xi\in X(\infty)$,
• $\{\mu_p\colon p\in X\}$ is $\Gamma$-invariant, that is, for every Borel set $A\subset X(\infty)$ and $\gamma\in\Gamma$,

$$\mu_{\gamma p}(\gamma A)=\mu_p(A).$$

Such measure can be constructed as follows. For Fuchsian groups, this construction is due to [Patterson '76]. For general hyperbolic spaces, this was investigated by [Sullivan '79]. The principle idea also extends to the construction of conformal measures in real and complex dynamics (see, for example, [Denker, Urbanski '91]) and has many variants.

Lemma 1. If $\Gamma$ is not of divergence type, then there exists a positive monotone increasing function $f\colon\mathbb R^+\to\mathbb R^+$ such that for every $a$ it holds $$\frac{f(r+a)}{f(r)}\to1 \quad\text{ as }r\to\infty$$ and the modified series $$\tilde P(s,x,y) := \sum_{\gamma\in\Gamma}f(d(x,\gamma x))e^{-sd(x,\gamma y)}$$ is of divergence type.

By the above, up to modifying the Poincaré series, we can assume that $\Gamma$ is of divergence type. Given $x\in X$, $s>\delta$, and $p\in X$, consider the measure $$\mu_{p,x,s} := \frac{\sum_{\gamma\in\Gamma}e^{-sd(p,\gamma x)}\delta_{\gamma x}} {P(s,x,x)},$$ where $\delta_y$ denotes the Dirac measure at $y$.

Lemma 2. It holds $$e^{-sd(p,x)} \le \mu_{p,x,s}(X\cup X(\infty)) \le e^{sd(p,x)}.$$ In particular, the measure is finite.

Proof. As $d(p,\gamma x)\le d(p,x)+d(x,\gamma x)$, it follows $$\mu_{p,x,s}(X\cup X(\infty)) = \frac{\sum_{\gamma\in\Gamma}e^{-sd(p,\gamma x)}} {\sum_{\gamma\in\Gamma}e^{-sd(x,\gamma x)}} \ge \frac{\sum_{\gamma\in\Gamma}e^{-s(d(x,\gamma x)+d(p,x))}} {\sum_{\gamma\in\Gamma}e^{-sd(x,\gamma x)}} \ge e^{-sd(p,x)}$$ together with the analogous upper bound.

Lemma 3. $\Gamma x\subset{\rm supp}\mu_{p,x,s}\subset\overline{\Gamma x}$.

Fix $p\in X$, choose $s_k\searrow\delta(\Gamma)$ as $k\to\infty$ and consider a weak$\ast$ limit $$\mu_p := \lim_{k\to\infty}\mu_{p,x,s_k}.$$ A priori, the limit measure may depend on the subsequence $(s_k)_k$.

Lemma 4. For every $p\in X$, the weak limit $\mu_p:=\lim_{k\to\infty}\mu_{p,x,s_k}$ exists. The family of measures $\{\mu_p\colon p\in X\}$ is a $\delta(\Gamma)$-dimensional Busemann density.

The Busemann density is also called Patterson-Sullivan measure.

Proof. Let us first show that $\mu_p$ is a Busemann density. Because is of divergence type and $\Gamma$ is discrete, it immediately follows that ${\rm supp}\mu_p\subset\overline{\Gamma x}\,\cap\, X(\infty)$. Indeed, for every bounded set $A\subset X$ it holds $\lim_{k\to\infty}\mu_{p,x,s_k}(A)=0$ and hence ${\rm supp}\mu_p\subset X(\infty)$.

By the above, for every $\xi\in X(\infty)$ there is $(\gamma_n)_n\subset\Gamma$ such that for every $x\in X$ it holds $\xi=\lim_n\gamma_nx$. We will use the following claim.

Claim. For every $p,q\in X$, it holds $\lim_n(d(q,\gamma_nx)-d(p,\gamma_nx))=b_p(q,\xi)$.

Given $p,q\in X$, by this claim, for any $s$ it holds $$\frac{e^{-sd(q,\gamma_n x)}}{e^{-sd(p,\gamma_n x)}} = e^{-s\big(d(q,\gamma_n x)-d(p,\gamma_n x)\big)} \to e^{-sb_p(q,\xi)}$$ as $n\to\infty$. As $\Gamma$ is discrete cocompact, $$a := \frac12\inf\{d(y,\gamma y)\colon\gamma\in\Gamma,y\in X\} >0.$$ For every $n\in\mathbb N$, $$\frac{\mu_{p,x,s}}{\mu_{q,x,s}}(B_a(\gamma_n x)) = \frac{e^{-sd(q,\gamma_n x)}}{e^{-sd(p,\gamma_n x)}} \to e^{-sb_p(q,\xi)}$$ as $n\to\infty$. Hence $$\frac{\mu_{p}}{\mu_{q}}(\xi) =\lim_{s_k\to\delta(\Gamma)}\lim_{n\to\infty} \frac{\mu_{p,x,s_k}}{\mu_{q,x,s_k}}(B_a(\gamma_n x)) = \lim_{s_k\to\delta(\Gamma)}e^{-s_kb_p(q,\xi)} = e^{-\delta(\Gamma) b_p(q,\xi)}.$$

Finally, let us show invariance. For every $s>\delta(\Gamma)$ and $\beta\in\Gamma$, using the fact that $\Gamma$ is a group of isometries, it holds $$\mu_{\beta p,x,s} = \frac{\sum_{\gamma\in\Gamma}e^{-sd(\beta p,\gamma x)}\delta_{\gamma x}} {P(s,x,x)} = \frac{\sum_{\gamma\in\Gamma}e^{-sd(\beta p,\beta\gamma x)}\delta_{\beta\gamma x}} {P(s,x,x)} = \frac{\sum_{\gamma\in\Gamma}e^{-sd( p,\gamma x)}\delta_{\beta\gamma x}} {P(s,x,x)}.$$ Hence, for every Borel measurable set $B\subset X\cup X(\infty)$, it holds $$\mu_{\beta p,x,s}(\beta B) = \mu_{p,x,s}(B).$$ Hence, for every Borel measurable set $A\subset X(\infty)$, it follows $\mu_p(A)=\mu_{\beta p}(\beta A)$, proving invariance. This shows all properties of a Busemann density.

It remains to show uniqueness (see [Knieper '97] for details).

We state further properties, which we will not show here (see [Knieper '97]).

Lemma. For every $p\in X$, ${\rm supp} \mu_p=X(\infty)$.

Denote by ${\rm pr}\colon \overline X\times X\setminus D\to X(\infty)$ the projection ${\rm pr}_y(x)$, which for every $x\in \overline X$ and $y\in X$, $y\ne x$, assigns $${\rm pr}_y(x):=\sigma_{y,x}(\infty).$$ Denote by $S_{d(y,x)}(y)$ the sphere of radius $d(y,x)$ and center $y$. For $\xi:= {\rm pr}_y(x)$ let $$B^\rho_r(\xi) :={\rm pr}_y(B_\rho(x)\cap S_{d(y,x)}(y)) \subset X(\infty)$$ a ball of radius $r=e^{-d(y,x)}$ at infinity, that is, at $\xi$.

Lemma. There exists $R>0$ and for every $\rho\ge 2R$ there is $b(\rho)>0$ such that for every $\xi\in X(\infty)$ it holds $$\frac{1}{b(\rho)}r^{\delta(\Gamma)} \le \mu_y(B_r^\rho(\xi)) \le b(\rho)r^{\delta(\Gamma)}.$$

In other words, relative to the conformal structure on $X(\infty)$, $\mu_y$ behaves like an $\delta(\Gamma)$-dimensional Hausdorff measure. More precisely, the $\delta$-dimensional Busemann density is Lipschitz equivalent to the $\delta$-dimensional Hausdorff measure.

Uniqueness (up to some constant) of the Busemann densities $\{\mu_p\}$ for $X/\Gamma$ a compact rank 1 manifold is stated in [Knieper '97]. The itens of proof are: Every Busemann density is $\delta(\Gamma)$-dimensional and Lipschitz equivalent to the $\delta(\Gamma)$-dimensional Hausdorff measure on $X(\infty)$. Moreover, it is invariant and ergodic with respect to the induced action of $\Gamma$ on $X(\infty)$. This implies that any pair of Busemann densities agree up to some constant.