We study valued fields equipped with an automorphism σ which is
locally infinitely contracting in the sense that α≪σα for
all 0<α∈Γ. We show that various notions of valuation theory, such
as Henselian and strictly Henselian hulls, admit meaningful transformal
analogues. We prove canonical amalgamation results, and exhibit the way that
transformal wild ramification is controlled by torsors over generalized vector
groups. Model theoretically, we determine the model companion: it is decidable,
admits a simple axiomatization, and enjoys elimination of quantifiers up to
algebraically bounded quantifiers.
The model companion is shown to agree with the limit theory of the Frobenius
action on an algebraically closed and nontrivially valued field. This opens the
way to a motivic intersection theory for difference varieties that was
previously available only in characteristic zero. As a first consequence, the
class of algebraically closed valued fields equipped with a distinguished
Frobenius x↦xq is decidable, uniformly in q.Comment: identical to v1 apart from slight modifications in abstrac