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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) ?

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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.

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  • $\begingroup$ Are you refering to the Shannon–McMillan–Breiman theorem ? Could you give a hint about how to apply it ? $\endgroup$
    – Stéphane Laurent
    Commented Jun 18, 2015 at 15:14
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    $\begingroup$ Actually just Shannon's version of this theorem (convergence in probability) is enough. Namely, let $h<\log\#A$ be the entropy of $(X_n)$. Then for any $\epsilon>0$ there are sets $C_n\subset A^n$ with the property that that their measure goes to 1, whereas $\#C_n\le e^{(h+\epsilon)n}$. Then for small enough $\epsilon$ the sets $B_n=A^n\setminus C_n$ fail to satisfy your condition. $\endgroup$
    – R W
    Commented Jun 18, 2015 at 17:33

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