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!



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