I have a smooth function on a manifold, $f:X\to X$, with the following property. Although $f$ is not injective, there exists an integer $N\geq1$ such that the restriction of $f$ to $f^N(X)$ is injective. Equivalently, for all $x,y\in X$, if $f^{N+1}(x)=f^{N+1}(y)$ then $f^N(x)=f^N(y)$. Is there a word for such an $f$ in the literature?
I'm basically looking for a term that communicates, "From the point of view of the theory of dynamical systems, $f$ might as well be injective; I'm just too lazy to restrict its domain manually".
I tried Googling "X injective" for adverbs $\mathrm X\in\{\mathrm{eventually}, \mathrm{nearly}, \mathrm{almost}, \mathrm{quasi-}\}$ but those phrases all mean something else. Also "unipotent map" is taken, and neighboring phrases like "uniformly co-unipotent" don't get any hits.