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Let $\phi$ be a pseudo-Anosov of a compact oriented surface $F$ with boundary. Let $\beta\subset F$ be a simple closed loop and $\alpha$ either a simple closed loop or an embedded arc with endpoints in $\partial F$. Is it true that $$lim_{n\to \infty}\, \frac{i(\phi^n(\alpha),\beta)}{\lambda^n}=I(\alpha,\cal F^s,\mu^s)\cdot I(\beta, \cal F^u,\mu^u),$$ where $i$ is the geometric intersection number, $I$ its generalization to measured foliations, $\lambda=$ the strech factor, and $(F^s,\mu^s)$ and $(F^u,\mu^u)$ the stable and unstable measured foliations of $\phi$? (Where $\mu^s\otimes \mu^u(F)=1.$)

That was proved for closed surfaces by Thurston, cf. "Thurston's Work on Surfaces" by Fathi, Laudenbach, Poenaru, translated by Kim and Margalit. However the proof is non trivial, so I am curious in particular if the closed surface case can be somehow bootstrapped to the bordered case. (Or perhaps there is even some mention of it in the original Asterisque article, which I have no access to.)

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