Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model.
Suppose also that the angle between these axes is small and let $W$ be the geodesic which bisects this angle, and with the same orientation. Can we say anything about how a point progresses along $W$ as we hit it with $X$ and $Y$?
This picture should clarify a bit the situation:
I would like to say that $d_(\pi_W(e_1\ldots e_k O),\pi_W(e_1\ldots e_k e_{k+1}O))>\varepsilon$, where $e_i=X,Y$ and $\pi_W$ is the nearent point projection onto the geodesic $W$.