= Finite one-sided generator of an invertible system forces zero entropy
If an invertible probability <measure-preserving system> has a finite <one-sided generator> $\xi$, then its future sigma-algebra $\mathcal F_1=T^{-1}\mathcal B$ equals $\mathcal B$ modulo null sets. Hence $H_\mu(\xi\mid\mathcal F_1)=0$, and the <infinite-future formula for partition entropy rate> and <Kolmogorov-Sinai generator theorem> give $h_\mu(T)=0$. Invertibility matters: a fair binary one-sided <Bernoulli shift> has entropy $\log2$ and a finite one-sided generator.
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