We introduce the notion of a twisted differential operator of given radius relative to an
endomorphism
σ of
an affinoid algebra A.
We show that this notion is essentially independent of the choice of the
endomorphism σ.
As a particular case, we obtain an explicit equivalence between modules endowed with a
usual integrable connection (i.e., differential systems) and modules endowed with a
σ-connection
of the same radius (this concept generalizes both finite difference and
q-difference
systems). Moreover, this equivalence preserves cohomology and in particular
solutions.