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--- ebsd2021:tema12 [2021/09/22 10:19] – created escola
+++ ebsd2021:tema12 [2021/09/22 10:24] (current) – escola
@@ Line 1: / Line 1: @@
+**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:
-  * $\supp\mu\subset\Lambda(\Gamma)$,
+  * ${\rm supp}\mu\subset\Lambda(\Gamma)$,
   * $\frac{d\mu_p}{d\mu_q}(\xi)=e^{-\alpha b_p(q,\xi)}$ 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 $\gamma\in\Gamma$,
@@ Line 10: / Line 12: @@
 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 $f\colon\bR^+\to\bR^+$ such that for every $a$ it holds
+**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
@@ Line 18: / Line 20: @@
 \[
 	\tilde P(s,x,y)
-	\eqdef \sum_{\gamma\in\Gamma}f(d(x,\gamma x))e^{-sd(x,\gamma y)}
+	:= \sum_{\gamma\in\Gamma}f(d(x,\gamma x))e^{-sd(x,\gamma y)}
 \]
 is of divergence type.
@@ Line 38: / Line 40: @@
 In particular, the measure is finite.
-**Proof.** As
+**Proof.** As $d(p,\gamma x)\le d(p,x)+d(x,\gamma x)$, it follows
-$
-	d(p,\gamma x)\le d(p,x)+d(x,\gamma x),
-$
-it follows
 \[
 	\mu_{p,x,s}(X\cup X(\infty))
@@ Line 53: / Line 51: @@
 together with the analogous upper bound.
-**Lemma 4.** 	$\Gamma x\subset\supp\mu_{p,x,s}\subset\overline{\Gamma x}$.
+**Lemma 3.** $\Gamma x\subset{\rm supp}\mu_{p,x,s}\subset\overline{\Gamma x}$.
@@ Line 63: / Line 61: @@
 A priori, the limit measure may depend on the subsequence $(s_k)_k$.
-**Lemma 5.** 	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.
+**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//.
@@ Line 122: / Line 120: @@
 We state further properties, which we will not show here (see [Knieper '97]).
-**Lemma.** For every $p\in X$, $\supp \mu_p=X(\infty)$.
+**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 $\pr_y(x)$, which for every $x\in \overline X$ and $y\in X$, $y\ne x$, assigns
+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 $\xi:= \pr_y(x)$ let
+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)