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How to prove the following lemma from the book "Closed curves on surfaces" written by Francis Bonahon?

Lemma: For any fattened train track $\Phi$, the number of edge paths of $\Phi$ of length $r$ that are followed by embedded arcs grows polynomially with $r$.

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migrated from mathoverflow.net Dec 30 '13 at 19:01

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    $\begingroup$ If anyone wanted to search MO for this question using the words in the lemma they'd be thwarted by the use of an image instead of text. $\endgroup$ – Dan Piponi Dec 30 '13 at 16:04
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    $\begingroup$ It would be very nice to make the question self contained. What is a "fattened train track"? What is an "edge path"? $\endgroup$ – Daniel Soltész Dec 30 '13 at 16:15
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    $\begingroup$ This lemma is at the bottom of page 22: www-bcf.usc.edu/~fbonahon/Research/Preprints/Bouquin.pdf The proof is on page 23. What's the problem? The basic idea is that because the arcs are embedded, you only get a polynomial number of choices on each piece (each switch of the train track). $\endgroup$ – Douglas Zare Dec 30 '13 at 16:46
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This is a very nice and very fundamental exercise. Here are many hints to guide you through it.

Step one: Introduce coordinates. If $\alpha$ is such an arc, then it runs across each branch (edge) of the fattened train track some number of times. This gives you a number for each branch. If $\alpha$ has length $r$ then the sum of these numbers is $r$.

Step two: These coordinates determine the arc $\alpha$ up to isotopy. So if $\alpha$ and $\beta$ have the same coordinates, then they are isotopic. (This is the step that fails for homotopy classes of immersed arcs.)

$\newcommand{\ZZ}{\mathbb{Z}}$ Step three: These coordinates are integral, so they give an injective map of isotopy classes of arcs to $\ZZ^B$, where $B$ is the set of branches.

Step four: Count lattice points inside the cube of radius $r$. Done.

Further exercise: Give a better upper bound on the degree of the polynomial.

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