joining or coupling

Question

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

Answer 1

I assume that the context you're talking about is ergodic measures on $A^{\mathbb Z}$ for some finite set $A$. One definition of the $\bar d$ distance is $$ \bar d(\mu,\nu)=\inf_{\lambda\in J(\mu,\nu)}\int \mathbb 1_{x_0\ne y_0} \,d\lambda(x,y), $$ where $J(\mu,\nu)$ is the (weak$^*$-compact) collection of joinings of $\mu$ and $\nu$, that is the collection of shift-invariant measures, $\lambda$, on $A^{\mathbb Z}\times A^{\mathbb Z}$ such that $\lambda(B\times A^{\mathbb Z}=\mu(B)$ and $\lambda(A^{\mathbb Z}\times B)=\nu(B)$.

I have a paper that estimates $\bar d$ distance for infinite order Markov processes with Zaqueu Coelho. Xavier Bressaud, Roberto Fernandez and Antonio Galves have a paper dealing with a similar situation.