Examples of ergodic processes uniquely determined by their two-marginal laws (Q5951985)

Let \(\sigma\) be a shift on \(\Omega=\{0,\dots,k-1\}^{\mathbb Z}\). Consider a \(\sigma\)-invariant probability measure \(\mu\) on \(\Omega\). Let \(C_{\mu}\) be the family of all \(\sigma\)-invariant probability measures \(\nu\) on \(\Omega\) having the two-dimensional marginals \(\nu(\{\omega\in\Omega:\omega_0=i, \omega_n=j\}) = \mu(\{\omega\in\Omega:\omega_0=i, \omega_n=j\})\). Assume that \((\sigma,\mu)\) is ergodic. Then the sub-family of all \(\nu\in C_{\mu}\) for which \(\sigma\) is ergodic is denoted by \(E_{\mu}\). The authors give two classes of ergodic \(\mu\) for which \(E_{\mu}\) is reduced to \(\{\mu\}\) when \(k=2\). One class corresponds to a partition of the unit circle under an irrational rotation, the other class is associated with a Kronecker Gaussian dynamical system.