Attractive nearest-neighbor spin systems on the integers in a randomly evolving environment

We consider spin systems on $\Z$ (i.e.\ interacting particle systems on $\Z$ in which each coordinate only has two possible values and only one coordinate changes in each transition) whose rates are determined by another process, called a background process. A canonical example is the so called contact process in randomly evolving environment (CPREE), introduced and analysed by E. Broman and furthermore studied by J. Steif and the author, where the marginals of the background process independently evolve as 2-state Markov chains and determine the recovery rates for a contact process. We prove that under certain conditions on the rates there are at most two extremal stationary distributions. The proof follows closely the ideas of Liggett's proof of a corresponding theorem for spin systems on $\Z$ without a background process.