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 $r$.
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.