Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$. Consider $0<\alpha<1$ and let $$K_\alpha=\{(x_i)\in\Omega:\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}x_i=\alpha\}.$$ Let $\mathcal{M}_\sigma(\Omega)$ stand for the family of all shift invariant Borel probability measures on $\Omega$. For $\mu\in\mathcal{M}_\sigma(\Omega)$ we write $h(\mu)$ for the Kolmogorov-Sinai (metric) entropy of $\mu$. Let $M_\alpha$ be the set of shift invaraint measures concentrated on $K_\alpha$, that is, $M_\alpha= \{\mu\in\mathcal{M}_\sigma(\Omega):\mu(K_\alpha)=1\}$. It is easy to see that $M_\alpha$ is a closed subset of $\mathcal{M}_\sigma(\Omega)$ equipped with the weak$^*$ topology. What can be said about the number $\eta=\sup\{h(\mu): \mu\in M_\alpha\}$?

It is clear that the supremum is achieved by some ergodic measure, because $\mu\mapsto h(\mu)$ is upper semicontinous on $\mathcal{M}_\sigma(\Omega)$. But is a measure achieving that maximum unique?

Uniqueness is true for $\alpha=1/2$, where the Bernoulli measure attains the maximum.

A similar (but I am not sure if equivalent question) is the following:

Let $K_\alpha'$ be the set of all numbers in the unit interval whose binary expansion belongs to $K_\alpha$. What is the Hausdorff dimension of $K'_\alpha$?