Differences

This shows you the differences between two versions of the page.

--- ebsd2021:bourdon [2021/10/17 10:21] – created escola
+++ ebsd2021:bourdon [2021/10/18 10:53] (current) – tahzibi
@@ Line 2: / Line 2: @@
-We consider the \emph{Gromov product} of $a,b\in X$ relative to $x\in X$,
+We consider the //Gromov product// of $a,b\in X$ relative to $x\in X$,
 \[
 	(a,b)_x
-	\eqdef\frac12\big(d(x,a)+d(x,b)-d(a,b)\big).
+	:=\frac12\big(d(x,a)+d(x,b)-d(a,b)\big).
 \]
-Based on the Busemann function, consider the \emph{horospherical distance} (relative to $\xi\in X(\infty)$), given a geodesic ray $\gamma$ with $\gamma(0)=y$ and $\gamma(\infty)=\xi$,
+Based on the Busemann function, consider the //horospherical distance// (relative to $\xi\in X(\infty)$), given a geodesic ray $\gamma$ with $\gamma(0)=y$ and $\gamma(\infty)=\xi$,
 \[
 	\beta_\xi(x,y)
-	\eqdef b_y(x,\xi)
+	:= b_y(x,\xi)
 	= \lim_{t\to\infty}d(x,\gamma(t))-d(y,\gamma(t)).
 \]
@@ Line 17: / Line 17: @@
 \[
 	(\xi,\eta)_x
-	\eqdef \lim_{n\to\infty}(a_n,b_n)_x
+	:= \lim_{n\to\infty}(a_n,b_n)_x
 \]
 (this limit exists and is independent on the chosen sequences, see above).
@@ Line 31: / Line 31: @@
 \end{split}\]
-\begin{lemma}[{Bourdon metric \cite[Chapter III.H Propositions 3.7 and 3.21]{BriHae:99}, \cite[Théorème 2.5.1]{Bou:95}}]
+For the following see [Bridson, Haefliger, Chapter III.H Propositions 3.7 and 3.21] or [Bourdon 1995].
-	For any $x\in X$, the function
+**Lemma.** For any $x\in X$, the function
 \[
 	d_x(\xi,\eta)
-	\eqdef\begin{cases}
+	:=\begin{cases}
 		e^{-(\xi,\eta)_x}&\text{ if }\xi\ne\eta,\\
 &\text{ otherwise},
@@ Line 45: / Line 46: @@
 	= e^{\frac12(b_\xi(x,y)+b_\eta(x,y))}d_x(\xi,\eta).
 \]
-\end{lemma}
-This will be used some regularity properties of Gibbs cocycles.
+The above result will be used to state some regularity properties of Gibbs cocycles.