Let $(\cal F,\mu)$ be the stable measured foliation of a pseudo-Anosov on an oriented surface $S$. Can there be two non-isotopic multi-loops (collections of disjoint simple loops) $L\_1,L\_2\subset S$, both transverse to $\cal F$ with the same transverse measure, $\mu(L\_1)=\mu(L\_2)$?


If yes, how one can construct such example?


PS. One can prove that "transversely isotopic"="isotopic" in this setting.



Two multi-curves in a surface with the same transverse measure
Yes. For example, by lifting - there are other constructions, as well.




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Here are the details of a lifting construction. Suppose that $f$ is a pseudo-Anosov on a surface $T$. Pick a simple closed curve $\alpha$ in $T$. Choose a double cover $S$ of $T$ where the preimage of $\alpha$ has two components. 


There is some power $k \in \mathbb{N} - \{0\}$ so that $f^k\_\* \colon \pi\_1(T) \rightarrow \pi\_1(T)$ preserves the subgroup $\pi\_1(S)$. Thus $f^k$ lifts to $S$. Let $g$ be any one such lift; note that $g$ is pseudo-Anosov on $S$. 


Finally, the two lifts of $\alpha$ to $S$ have the desired property with respect to the stable measured foliation of $g$.


