Shnirman, M. G. Conditions for the nonergodicity of one-dimensional Stavskaya media. (English. Russian original) Zbl 0573.60061 Sel. Math. Sov. 4, 63-71 (1985); translation from Interacting Markov processes and their applications to biology, Pushchino 1979, 37-48 (1979). Consider a homogeneous Markov chain, with states \(x=(x_ h)\), which are mappings from \({\mathbb{Z}}\) to the finite set \(N=\{0,1,...,n\}\). It is determined by a (transition) operator P, assumed homogeneous, local, and with independent transitions. An operator P is called determinate if it maps every point measure to another point measure. Operators for which the measure \(\bar n \)(the measure concentrated on the state ”\(x_ h=n\) for all h”) is invariant are called Stavskaya operators. In this paper, the author considers Stavskaya operators P which are represented as \(P=SQ\), with Q determinate and S a noise operator (defined by: if \(x_ h=n\), then \((S_ x)_ h(n)=1\) and if \(x_ h\neq i\), then \((S_ x)_ h(i)<1)\). Such operators clearly have the property \(P\bar n=\bar n\), and the present paper provides conditions under which P is nonergodic, i.e. possesses invariant measures, different from \(\bar n.\) In particular, it does not assume that \(Q\bar O=\bar O\), or that Q be monotonic. Reviewer: W.Wreszinski MSC: 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60B99 Probability theory on algebraic and topological structures Keywords:nonergodicity; Stavskaya operators; noise operator; invariant measures PDFBibTeX XMLCite \textit{M. G. Shnirman}, Sel. Math. Sov. 4, 63--71 (1985; Zbl 0573.60061); translation from Interacting Markov processes and their applications to biology, Pushchino 1979, 37--48 (1979)