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--- ebsd2021:tema12 [2021/09/22 10:21] – 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:
   * ${\rm supp}\mu\subset\Lambda(\Gamma)$,
@@ Line 49: / Line 51: @@
 together with the analogous upper bound.
-**Lemma 4.** $\Gamma x\subset{\rm supp}\mu_{p,x,s}\subset\overline{\Gamma x}$.
+**Lemma 3.** $\Gamma x\subset{\rm supp}\mu_{p,x,s}\subset\overline{\Gamma x}$.
@@ Line 59: / 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 120: / Line 122: @@
 **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)