We approach the study of complete bifix decodings of (uniformly) recurrent
languages with the help of the free profinite monoid. We show that the complete
bifix decoding of a uniformly recurrent language F by an F-charged rational
complete bifix code is uniformly recurrent. An analogous result is obtained for
recurrent languages.Comment: Original Manuscript of article to be published by De Gruyter in Forum
Mathematicum. The last section of the version in Forum Mathematicum is very
different, as there it is not proved that the Sch\"utzenberger group is an
invariant of eventual conjugacy (the argument in the Original Manuscript had
a flaw), but only that its maximal pronilpotent quotient is invariant by
eventual conjugac