Dynamics on free-by-cyclic groups

Given a free-by-cyclic group $G=F_N{⋊}_{\phi }\mathbb{Z}$ determined by any outer automorphism $\phi \in \mathrm{O}\mathrm{u}\mathrm{t}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $$\mathcal{A}$ \subset H^1(X;$\mathbb{R}$) = $\mathrm{H}\mathrm{o}\mathrm{m}$(G;$\mathbb{R}$)$containingthehomomorphism$u_0 \colon G \to $\mathbb{Z}$$having$$\mathrm{k}\mathrm{e}\mathrm{r}$(u_0) = F_N$, a homology class $ϵ\in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak{H}\text{\hspace{0.17em}⁣}:\mathcal{A}\to \mathbb{R}$ with the following properties. Given any primitive integral class $u\in \mathcal{A}$ there is a graph ${\mathrm{\Theta }}_u\subset X$ such that: (1) the inclusion $\Theta_u \to X$is$\pi_1$-injectiveand$\pi_1(\Theta_u) = $\mathrm{k}\mathrm{e}\mathrm{r}$(u)$,(2)$u(\epsilon) = \chi(\Theta_u)$,(3)$\Theta_u \subset X$ is a section of the semiflow and the first return map to ${\mathrm{\Theta }}_u$ is an expanding irreducible train track map representing ${\phi }_u\in \mathrm{O}\mathrm{u}\mathrm{t}(\mathrm{k}\mathrm{e}\mathrm{r}(u))$ such that $G=\mathrm{k}\mathrm{e}\mathrm{r}(u){⋊}_{{\phi }_u}\mathbb{Z}$, (4) the logarithm of the stretch factor of ${\phi }_u$ is precisely $\mathfrak{H}(u)$, (5) if $\phi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism ${\phi }_u$ of $\mathrm{k}\mathrm{e}\mathrm{r}(u)$ is also hyperbolic and fully irreducible.