Let $(X,\mathcal B,\mu,T)$ be a probability-preserving system, meaning that $(X,\mathcal B,\mu)$ is a probability space and $T:X\to X$ is a measurable map with $\mu(T^{-1}A)=\mu(A)$ for every $A\in\mathcal B$. Let $\mathcal P$ be a finite measurable partition of $X$. For $n\ge 1$, set
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\begin{align*}
\mathcal P_0^{n-1}=\bigvee_{k=0}^{n-1}T^{-k}\mathcal P
\end{align*}
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and write $H_\mu(\mathcal Q)$ for the Shannon entropy of a finite measurable partition $\mathcal Q$. Then the limit