✎ ebsd2021:bourdon []

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Let $X$ be a complete simply connected Riemannian manifold of dimension $\ge2$ and pinched negative curvature, $ X(\infty)$ its boundary at infinity, $\Gamma$ a non-elemantary discrete group of isometries of $X$.

We consider the //Gromov product// of $a,b\in X$ relative to $x\in X$,
\[
	(a,b)_x
	:=\frac12\big(d(x,a)+d(x,b)-d(a,b)\big).
\]
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 $\gamma(\infty)=\xi$,
\[
	\beta_\xi(x,y)
	:= b_y(x,\xi)
	= \lim_{t\to\infty}d(x,\gamma(t))-d(y,\gamma(t)).
\]

Given $\xi,\eta\in X(\infty)$, $\xi\ne \eta$, and $a_n\to \xi, b_n\to \eta$, then
\[
	(\xi,\eta)_x
	:= \lim_{n\to\infty}(a_n,b_n)_x
\]
(this limit exists and is independent on the chosen sequences, see above).
If $\xi=\gamma(\infty)$ and $\eta=\gamma(-\infty)$ for a geodesic $\gamma$, given $z=\gamma(0)$ (or any other point in $\gamma$),
\[\begin{split}
	(\xi,\eta)_x
	&= \lim_n\,(\gamma(n),\gamma(-n))_x\\
	&= \lim_n\frac12\big(d(x,\gamma(n))+d(x,\gamma(-n))-d(\gamma(n),\gamma(-n)\big)\\
	&= \lim_n\frac12\big(d(x,\gamma(n))+d(x,\gamma(-n))-\Big(d(\gamma(n),z)+d(z,\gamma(-n)\Big)\big)\\
	&= \lim_n\frac12\big(d(x,\gamma(n))-d(z,\gamma(n))\big)+
	\lim_n\frac12\big(d(x,\gamma(-n))-d(z,\gamma(-n)\big)\\
	&= \frac12(b_\xi(x,z)+b_\eta(x,z)).
\end{split}\]
 
\begin{lemma}[{Bourdon metric \cite[Chapter III.H Propositions 3.7 and 3.21]{BriHae:99}, \cite[Théorème 2.5.1]{Bou:95}}]
	For any $x\in X$, the function
\[
	d_x(\xi,\eta)
	:=\begin{cases}
		e^{-(\xi,\eta)_x}&\text{ if }\xi\ne\eta,\\
		0&\text{ otherwise},
		\end{cases}
\]	
defines a metric on $ X(\infty)$ which induces the same topology as the cone topology. Moreover, for any $x,y\in X$
\[
	d_y(\xi,\eta)
	= e^{\frac12(b_\xi(x,y)+b_\eta(x,y))}d_x(\xi,\eta).
\]
\end{lemma}

This will be used some regularity properties of Gibbs cocycles.

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