  We prove that the class of separably algebraically closed valued fields
equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is
decidable, uniformly in $q$. The result is a simultaneous generalization of the
work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and
the work of the first author and Hrushovski (in the case where the fields are
algebraically closed).
  The logical setting for the proof is a model completeness result for valued
fields equipped with an endomorphism $\sigma$ which is locally infinitely
contracting and fails to be onto. Namely we prove the existence of a model
complete theory $\widetilde{\mathrm{VFE}}$ amalgamating the theories
$\mathrm{SCFE}$ and $\widetilde{\mathrm{VFA}}$ introduced in [4] and [9],
respectively. In characteristic zero, we also prove that
$\widetilde{\mathrm{VFE}}$ is NTP$_2$ and classify the stationary types: they
are precisely those orthogonal to the fixed field and the valuation group.
