Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is *asymptotically full* if for every increasing sequence of subsets $B_n \subset A^n$ such that $\dfrac{\#B_n}{\#A^n} \to 1$, one has
$$\Pr\bigl[(X_1, \ldots, X_n) \in B_n)\bigr] \to 1.$$ In other words, the law of $(X_1, \ldots, X_n)$ is asymptotically equivalent to the counting measure on $A^n$.

Now, say that a dynamical system $T$ is *nice* if it admits a generating partition such that the associated stationary process of names is asymptotically full.

Obviously, a Bernoulli automorphism $T$ is nice. I have not checked the case of an irrational rotation, but I think it is not nice.

Is there a relation between nice and positive entropy ? Or another relation with an other invariant property of dynamical systems (e.g. about the spectrum) ?