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The entropy rate of $\xi$ isThe numerator is a subadditive sequence, so the limit exists and equals the infimum of the displayed ratios.

For a finite measurable partition $\xi$,Its conditional entropy of finite measurable partitions relative to $\eta$ isThe concavity of $-t\log t$, equivalently conditioning reduces entropy, gives

The atoms of $T^{-1}\xi$ are $T^{-1}A$ with the same measures as the atoms $A$ of $\xi$. Hence $\boxed{H_{\mu }(T^{-1}\xi )=H_{\mu }(\xi )}$.

SetThe chain rule and invariance give $a_{N+M}\le a_N+a_M$, so $(a_N)$ is a subadditive sequence. ThereforeThe Kolmogorov-Sinai entropy is $h_{\mu }(T)={\sup }_{\xi }{h}_{\mu }(T,\xi )$ over finite partitions.

Taking $F=\{0,\dots ,N-1\}$ immediately shows that the infimum over arbitrary finite $F$ is at most $h_{\mu }(T,\xi )$. For the reverse inequality, apply Shearer's inequality to translates of a fixed finite $F$ inside a long interval. Every interior coordinate is covered $|F|$ times, while only $O(\max F)$ boundary coordinates are lost. Subadditivity bounds the boundary contribution; division by the interval length and passage to the limit giveTaking the infimum proves

LetThe events $\{0\leftrightarrow e_n\}$ and $\{e_n\leftrightarrow e_{n+m}\}$ are increasing events of bond percolation. The FKG inequality and translation invariance giveThus $(a_n)$ is a subadditive sequence. Since every probability is positive for $p>0$, Fekete lemma applies and gives