Measures maximizing entropy in a set of measures with fixed average for some observable

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 $K'_\alpha$?

• A low-brow heuristic way of thinking about this (compared to Vaughn's comprehensive answer below) is to say think about a block of length $N$. Your constraint ensures there are roughly $\alpha N$ 1's in the block. You are then interested in maximizing entropy. This can be achieved by putting the uniform distribution on all configurations with $\alpha N$ 1's. – Anthony Quas May 17 '18 at 15:32

In the setting you describe, for each $\alpha \in (0,1)$ the $(1-\alpha,\alpha)$-Bernoulli measure is the unique measure achieving the maximum. The function $\alpha \mapsto \eta(\alpha)$ is the Legendre transform of the function $t\mapsto P(t\phi)$ where $\phi(x) = x_0$ and $P$ is topological pressure. This is all part of the "multifractal analysis" of the system: see this other question and my answer there for some references and some more explanation.
As for the Hausdorff dimension of $K_\alpha'$, it's given by $\eta /\log 2$, since $\log 2$ is the Lyapunov exponent of the doubling map $f$, and binary expansions are codings of trajectories under $f$. Informally this follows from the relationship dimension = entropy / exponent; to make this a little more precise you can consider Bowen's equation, which gives Hausdorff dimension of a set $E$ as the unique root of $t\mapsto P_E(t\phi)$, where $P_E$ is topological pressure, but this time defined for sets that need not be compact or invariant, using the definition of Pesin-Pitskel (following Bowen's noncompact entropy definition); see this other question for some more details and references.