Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the volume measure in the following meaning :

\begin{equation} \frac{1}{l(\gamma_n)} \int_0^{l(\gamma_n)} f(x(t))dt \underset{n \to \infty}{\longrightarrow} L_g(f) \end{equation}

for all continuous functions $ f : S \to \mathbb{R}$. ( $x(t)$ the length parametrization of $\gamma_n$)

On the other hand, thanks to the work of Thurston, we have the following : \begin{equation} \frac{i(\gamma_n, c)}{l(\gamma_n)} \underset{n \to \infty}{\longrightarrow} l(c) \end{equation} for all closed curve $c$, where $i(\alpha,\beta)$ is defined as the geometric intersection between the geodesics $\alpha$ and $\beta$. ($\gamma_n$ is still a sequences of closed geodesics approximating the volume measure in the previous meaning).

if we look at the sequence $ (\frac{i(\gamma_n, \cdot)}{l(\gamma_n)})$ as a sequence of functions from the set of curves to in $\mathbb{R}$ (this is the role of the intersection) we could say that this sequence converge weakly to the function length (another function from curves to $\mathbb{R}$) (which could be interpreted as the intersection between the volume measure and the curve).

My question is : could we expect more (for a particular choice of sequence $(\gamma_n)$ for instance) than a weak convergence for the sequence of functions $ (\frac{i(\gamma_n, \cdot)}{l(\gamma_n)} : \{ \text{curves} \} \to \mathbb{R})$ ? Thank you for your attention.