Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does not contain the identity, and for a pair of $g,g^{-1}\in \Gamma$ contains only one of them.

If a point in $\Lambda(\Gamma)$ is an attracting point for some element $g$ of $\Gamma'$, then, quite obviously, the Euclidean distance $|g^n(x)-g^n(y)|$ decreases exponentially fast.

However, what if a point is not an attractive point of any element, but, nevertheless, there exists a sequence $g_n$ such that $|g_n(x)-g_n(y)|$ converges to 0; how fast does this converge?

I assume this corresponds to the following geometric picture: if $x,y$ are repelling and attracting points of some element of the Schottky group $T_{x,y}$. Then I presume that the images of these points under $g_n$ correspond to non-trivial closed geodesics in the handle body that converge to some kind of foliation, and it would be nice to know how fast the numbers of leaves grows, or else,how well this point is approximated by the corresponding sequences.

Since we are talking about Diophantine approximations now, can the bound be expressed purely in terms of the Hausdorff dimension of the limit set?

EDIT: Upon closer inspection, I presume that the bound must again be something like exponential, because if $|g_n(x)-g_n(y)|$ converges to $0$, then the points must lie in smaller and smaller circles eventually. Moreover, these circles must be contained in the circles, corresponding to powers of one of the generators that must occur in $g_n$ an increasing number of times (more or less pigeonhole principle,) so we can even have a lower bound on the speed of convergence.



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