  We prove that ascending HNN extensions of free groups are word-hyperbolic if
and only if they have no Baumslag-Solitar subgroups. This extends the theorem
of Brinkmann that free-by-cyclic groups are word-hyperbolic if and only if they
have no free abelian subgroups of rank 2. The paper is split into two
independent parts:
  1) We study the dynamics of injective nonsurjective endomorphisms of free
groups. We prove a canonical structure theorem that initializes the development
of improved relative train tracks for endomorphisms; this structure theorem is
of independent interest since it makes many open questions about injective
endomorphisms tractable.
  2) As an application of the structure theorem, we are able to (relatively)
combine Brinkmann's theorem with our previous work and obtain the main result
stated above. In the final section, we further extend the result to HNN
extensions of free groups over free factors.
