  We the study of the monodromy of local systems with bounded ramification on a
punctured disc defined over a non-archimedean valued field of characteristic
zero. First, we construct the local Fourier transforms and we establish their
main properties. Also, we show that the Fourier transform of a "perverse sheaf"
with bounded ramification on the affine line, is again a "perverse sheaf" with
bounded ramification (we put that in quotes, because actually the language of
perversity is not used). These foundations are then used to exhibit a natural
break decomposition for local systems with bounded ramification, analogous to
the classical one for Galois representation.
  This is the paper that was previously posted with the title "Local monodromy
in non-archimedean analytic geometry -- II". This revised release contains a
new section, with an application to the problem of localization of the
determinant of cohomology.
