Some remarks concerning the individual ergodic theorem of information theory

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 $\xi^n=\xi\vee T\xi\vee\dots<T^{n-1}\xi$ 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.