A notion of synchronization of symbolic dynamics and a class of   $C^*$-algebras
  We discuss a synchronization property for subshifts, that we call $\lambda$-synchronization. Under an irreducibility assumption we associate to a $\lambda$-synchronizing subshift a simple and purely infinite $C^*$-algebra. 
Wolfgang Krieger and Kengo Matsumoto
math.DS math.OA