of $\phi : F \to F$. The latter terminology stems from the fact that the injective endomorphism $\phi$ can be topologically represented by a graph map on the rose whose topological mapping torus has a fundamental group isomorphic to $F*_\phi$ . Following this analogy, we shall call $F$ the [**fiber**]{} and $\phi$ the [**monodromy**]{} of the mapping torus. Finally, when $\phi : F \to F$ is an automorphism, we call $F*_\phi = F \rtimes_\phi \mathbb Z$ a [**free-by-cyclic group**]{}.

The following are the properties of monodromies that we will study. An endomorphism $\phi: F \to F$ is [**reducible**]{} if there is a free factorization $A_1 * \cdots * A_k * B $ of $F$ where $B$ may be trivial if $k \ge 2$, and a sequence of elements, $(x_i)_{i=1}^k$, in $F$ such that $\phi(A_i) \le x_i A_{i+1} x_i^{-1}$ where the indices are considered$\mod k$; the collection $\{A_1, \ldots, A_k \}$ is a [**$\boldsymbol{\phi}$-invariant proper free factor system**]{}. An endomorphism is [**irreducible**]{} if it is not reducible. It is [**atoroidal**]{} if there does not exist a nontrivial element $a \in F$, element $x \in F$, and integer $n \ge 1$ such that $\phi^n(a) = x a x^{-1}$, equivalently, $i_x \phi^n(a) = 