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?

  • $\begingroup$ I believe Mark Bell at UIUC has thought about this, see this paper. $\endgroup$
    – Neal
    Nov 10 '16 at 2:12
  • 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 Mosher
    Nov 10 '16 at 2:49

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 $\mathrm{SL}(2, \mathbb{Z})$.


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