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) ?



Asymptotically full stationary process
In view of Shannon's theorem your definition implies that the entropy of $(X\_n)$ is $\log\# A$, which is only possible if $(X\_n)$ is Bernoulli with the uniform base distribution.


