Geodesic
Some remarks concerning the individual ergodic theorem of information theory
Matematičeskie zametki, Tome 9 (1971) no. 1, pp. 93-103 
B. S. Pitskel'. Some remarks concerning the individual ergodic theorem of information theory. Matematičeskie zametki, Tome 9 (1971) no. 1, pp. 93-103. http://geodesic.mathdoc.fr/item/MZM_1971_9_1_a13/
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     author = {B. S. Pitskel'},
     title = {Some remarks concerning the individual ergodic theorem of information theory},
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     pages = {93--103},
     year = {1971},
     volume = {9},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_1_a13/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $(X,\mu,T)$ be an ergodic dynamic system and let $\xi=(C_1,C_2,\dots)$ be a discrete decomposition of $X$. Conditions are considered for the existence almost everywhere of $$ \lim_{n\to\infty}\frac1n|\log\mu(C_{\xi n}(x))|, $$ where $C_{\xi n}(x)$ is the element of the decomposition $\xi^n=\xi\vee T\xi\vee\dots containing $x$. It is proved that the condition $H(\xi)<\infty$ is close to being necessary. If $T$ is a Markov automorphism and $\xi$ is the decomposition into states, then the limit exists, even if $H(\xi)=\infty$, and is equal to the entropy of the chain.