  This paper is about almost reducibility of quasi-periodic cocycles with a
diophantine frequency which are sufficiently close to a constant. Generalizing
previous works by L.H.Eliasson, we show a strong version of almost reducibility
for analytic and Gevrey cocycles, that is to say, almost reducibility where the
change of variables is in an analytic or Gevrey class which is independent of
how close to a constant the initial cocycle is conjugated. This implies a
result of density, or quasi-density, of reducible cocycles near a constant.
Some algebraic structure can also be preserved, by doubling the period if
needed.
