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 $\log 2$ and a finite one-sided generator.