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Let $(X,d)$ be a metric space.

Let $(x_n)_{n\in\mathbb N}$ be a geodesic ray: $d(x_n,x_m)=\vert n-m\vert$. Is it true that, for all $y\in G$, the sequence $d(x_{n+1},y)-d(x_n,y)$ converges to 1 as $n$ goes to infinity ?

I am particularly interested in the case of $\delta$-hyperbolic spaces. A positive answer to the above quetion would imply that any geodesic ray converges to a Busemann function (or horofunction).

More generally, is anything known about Busemann functions on hyperbolic spaces ? In particular, how do the Busemann compactification relates to the visual boundary ? These two boundaries are the same for CAT(0) spaces, but need not be in general, as shown by the example $\mathbb Z\times \mathbb Z/2\mathbb Z$, with obvious generating set.

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You can find some information on Busemann functions on $\delta$-hyperbolic spaces in Ghys-de la Harpe (``Sur les groupes hyperboliques d'après M. Gromov''). If I remember correctly, they show that Gromov boundary can indeed be modeled by Busemann functions modulo globally bounded functions (at least in the proper case). I don't think that you can hope for more. –  Theo Buehler Dec 9 '10 at 11:50
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Yes. Let $a_n:=d(y,x_n)-|n|$. By the triangle inequality, we see that $(a_n)_{n\ge0}$ is non-increasing and bounding. Therefore it has a finite limit $a$.

Now if $n\ge0$, $d(y,x_{n+1})-d(y,x_n)=a_{n+1}-a_n+1\rightarrow a-a+1=1$.

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