Let $G = \{0,1\}^{\mathbb{N}} = \mathbb{Z}\_{2}^{\mathbb{N}}$ be the Bernoulli space of two symbols, let $\sigma$ be the shift map and $M(G)$ the set of $\sigma$-invariant probabilities. Let $\bar{d}$ be the distance defined here: [joining or coupling](https://mathoverflow.net/questions/215066/joining-or-coupling). I put a bar now. We already know that, in this context, we have
$$\bar{d}(\eta,\mu)= \lambda([0]\times[1]) + \lambda([1]\times[0]),$$
where $\lambda$ is the joining which achieves the infimum in [joining or coupling](https://mathoverflow.net/questions/215066/joining-or-coupling).
We already know too that the entropy function $h:M(G) \rightarrow [0,\log2]$ is $\bar{d}$-continuous. 


My question is now the converse: Is $\bar{d}$ h continuous (in the point $(\frac{1}{2},\frac{1}{2})$ the Bernoulli uniform probability? 


More than that, would we have some good estimative like: if $h(\eta) > (\log2 - \varepsilon)$ so $\bar{d}(\eta,(\frac{1}{2},\frac{1}{2}))<\varepsilon$?


Thanks a lot for your attention



entropy and d-bar: how do we estimate continuity?
I don't know bounds, but you can get the continuity that you want. The key point is that the Bernoulli $(\frac 12,\frac 12)$ measure, $\mu$, is *finitely determined*. This means that if another measure $\nu$ satisfies (1) $\nu$ is *weak$^\*$*-close to $\mu$ (that is $\mu$ and $\nu$ are close on cylinder sets of length $n$) and (2) $h(\nu)>h(\mu)-\epsilon$, then $\nu$ is $\bar d$-close to $\mu$. The condition that $h(\nu)>\log 2-\epsilon$ ensures that on $n$-cylinders, $\nu$ is close to uniform; and then the result you want follows from the finitely determined property. 


See <http://www.scholarpedia.org/article/Ornstein_theory#Finitely_determined_.28FD.29_systems> for more details.


