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. 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. 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