Decompositions of factor maps involving bi-closing maps (Q1383740)

Consider a factor map \(\theta\) from an irreducible shift of finite type \(\sigma\) onto a sofic shift \(S\). The main result of the paper is the existence of a decomposition into factor maps \(\theta= \gamma\circ \phi\) where \(\phi\) maps onto another shift of finite type, and \(\gamma\) is bi-closing. Recall that a factor map is bi-closing if it never identifies two points which are asymptotic by the shift, in either direction. The proof is constructive. In general, it is impossible to postulate that the degrees of \(\theta\) and \(\gamma\) should be equal. Necessary and sufficient conditions for that are given.