given two shift invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$, is there a way to construct joinings of them? It's very diffcult, in general, to find exactly the minimal joining i.e, the joining which achieves the d-bar distance of the measures. I would like a way to construct joinings efficiently... The shift here is the usual full shift and the d-bar distance is

$d(\eta,\nu) = \inf_{\mu \in J}\mu([0] \times [1]) + \mu([1] \times [0])$

where $J$ is the space of joinings of $\mu$ and $\nu$. A Joining is a invariant measure in $\{0,1\}^{\mathbb{N}} \times \{0,1\}^{\mathbb{N}}$.which projects $\eta$ in the first coordinate and $\nu$ in the second... (Anthony Quas has defined this distance above, equal as my definition)

thanks for your attention