# Transition matrix associated to representative of element in $Out(F_n)$

There is a notion of transition matrix associated to elements in $$Out(F_n)$$ from Bestvina and Handel's paper that I am a little bit confused.

Let $$\Phi\in Out(F_n)$$ and $$\phi:\Gamma\to\Gamma$$ a homotopy equivalence being its representative, i.e. $$\Gamma$$ is a graph ($$n$$-rose) and $$\phi_*=\Phi$$. Let $$e_1,\ldots,e_n$$ be unoriented edges of $$\Gamma$$. Now, we can define the transition matrix $$M$$ as a $$n\times n$$ matrix whose entry $$a_{ij}$$ represents the number of times $$\phi(e_j)$$ crosses $$e_i$$. If this matrix is irreducible (and itself non-negative integered), by Perron-Frobinus we have a maximal eigenvalue $$\lambda$$ and one non-negative eigenvector which helps us to find a train track map of slope $$\lambda$$.

For example, consider $$\Phi\in Out(F_n)$$ such that $$a\mapsto b,b\mapsto c, c\mapsto d, d\mapsto a^{-1}d^{-1}c^{-1}b^{-1}$$. The transition matrix is $$M=\left[\begin{array}{l}0&0&0&1\\1&0&0&1\\0&1&0&1\\0&0&1&1\end{array}\right]$$ We now look for a metric encoded as $$v = [\ell(a),\ell(b),\ell(c),\ell(d)]^T$$ such that $$Mv=\lambda v$$.

Here is my first question: 1. The matrix multiplication gives us $$\ell(d)=\lambda\ell(a)$$ which is not quite right. We are looking for $$\ell(b)=\lambda\ell(a)$$ which is $$M^Tv=\lambda v$$. I am a little bit confused here.

My second question comes from the paper saying: "the ij-th entry of M^k equals the number of paths in $$\Gamma$$ that originates at $$j$$ terminates at $$i$$ crosses exactly $$k$$ edges". However, we only label the edges but not the vertices. From the above notation, what I have is for example, $$(1,1)$$-entry of $$M^2$$ equals $$\sum_{i\in\{a,b,c,d\}}\#(\phi(i) \text{crosses } a)\cdot \#(\phi(a) \text{crosses } i).$$ where I have trouble seeing the equivalence.

I am sure that if I carefully read the whole paper I'd find the answer. But I'd be appreciated if someone can elaborate it a little bit!