A class of nonmixing dynamical systems with monotonic semigroup property (Q810658)

Let \(k\geq 2\), and \(K=\{0,1,...,k-1\}\). Let \(\Pi\) be a strictly positive stochastic matrix for the state space K, and \(\mu_{\pi}\) the unique stationary Markov measure on \(\Omega =K^{{\mathbb{Z}}}\). The authors give a construction of stationary measures \(\nu\) on \(\Omega\) having the same two-dimensional marginal distributions as \(\mu_{\pi}\), and hence satisfying a Chapman-Kolmogorov equation, which are not Markovian. (Other examples of non-Markovian transition probabilities satisfying the Chapman-Kolmogorov equation have already been given by Lévy and Feller.) The authors assert that the shift under their measures \(\nu\) is ergodic but not mixing, but this is left for a separate note.