= Positive-word growth under a free-group endomorphism
Let an endomorphism of the <free group> on $x,y$ fix $x$ and send $y$ to $yw$, where $w$ is a nonempty word containing only the positive letters $x,y$. Every iterate of $y$ is again a positive word containing $y$. Substitution increases its length by $|w|$ times the number of occurrences of $y$, hence by at least $|w|$. No cancellation can occur, so $|\phi^n(y)|\ge1+n|w|$.
The uniqueness of the <reduced word in a free group> now proves that no positive iterate of the endomorphism is the identity. This criterion does not require the endomorphism to be invertible; for example the replacement $y\mapsto y^2$ is included.
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