Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance $n$ (with the word metric). Consider the probability measures $\mu_n$ and $\nu_n$ defined as $$ \mu_n := \frac{1}{|B(1,n)|}\sum_{\gamma\, \in B(1,n)}{\delta_{\gamma}} $$ and $$ \nu_n := \frac{1}{|\partial B(1,n)|}\sum_{\gamma\,\in\partial B(1,n)}{\delta_{\gamma}}. $$ It is clear that these measures converge weakly (there is a convergent sub-sequence) on the compact space $\Gamma\cup\partial\Gamma$. Moreover, the limit measure is supported on $\partial\Gamma$.
My question isare:
- Does anybody has have study the limit of these measuresas $n\to\infty$? Is there any type of (weak) convergence?
- Are these measures in some way related with the harmonic Hausdorff measure on the boundary$\partial\Gamma$ of $\Gamma$ ? to the Patterson-Sullivan measure? to the harmonic measure?
