One-dimensional Markov random fields, Markov chains and topological Markov fields (Q2862186)

The authors investigate the concept of topological Markov fields (TMF) in one dimension that is a new property of a shift space. Some relations beetwen a TMF and other notions such as Markov chain, topological Markov chain (TMC), Markov random field (MRF) and sofic shift are established. One of the main results is as follows:NEWLINENEWLINE Let \(X\) be a shift space. Then the following statements are equivalent:NEWLINENEWLINE (1) \(X\) is the support of a stationary Markov chainNEWLINENEWLINE (2) \(X\) is the support of a stationary MRFNEWLINENEWLINE(3) \(X\) is non-wandering and a TMFNEWLINENEWLINE(4) \(X\) is a TMC consisting of a finite union of irreducible TMCs with disjoint alphabets.NEWLINENEWLINE As a consequence, the authors deduce that stationary MRF is a Markov chain.NEWLINENEWLINEAnother main result is a finite procedure for checking whether a given sofic shift is a TMF where the input data is a labeled finite directed graph-presentation of a shift.