Let $(X,\mathcal B,\mu,T)$ and $(Y,\mathcal C,\nu,S)$ be probability-preserving dynamical systems, and let $\pi:X\to Y$ be a factor map, meaning that $\pi$ is $(\mathcal B,\mathcal C)$-measurable, $\pi_\#\mu=\nu$, and
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\begin{align*}
\pi\circ T = S\circ \pi
\end{align*}
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$\mu$-almost everywhere. Then the Kolmogorov-Sinai entropy of the factor system is bounded above by the entropy of the original system: