A profinite approach to complete bifix decodings of recurrent languages

Abstract

We approach the study of complete bifix decodings of (uniformly) recurrent languages with the help of free profinite monoids. 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. As an application of the machinery developed within this approach, we show that the maximal pronilpotent quotient of the Schützenberger group of an irreducible symbolic dynamical system is an invariant of eventual conjugacy.