The simplest case of a well known theorem of Penner states that given a pair of filling curves, a positive twist about one curve together with a negative twist about the other curve is a pseudo-anosov mapping class. Suppose I have an explicit pair of curves that fill a closed surface, is there a reasonable way to explicitly compute the stable and unstable laminations of this map?
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$\begingroup$ I believe Mark Bell at UIUC has thought about this, see this paper. $\endgroup$– NealNov 10, 2016 at 2:12
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3$\begingroup$ Penner's proof gives a very explicit description, in terms of a specific train track and specific Perron-Frobenius matrix on the branches of that train track, the eigenvector of which gives weights on those branches that determine the stable lamination. Similarly for the unstable. $\endgroup$– Lee MosherNov 10, 2016 at 2:49
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Your question is explicitly answered in Section 6 of Thurston's Bulletin article, freely available here. In the end it boils down to a calculation in $\mathrm{SL}(2, \mathbb{Z})$.
