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Corollary 4.14.3 in Walters' book states that every zero-entropy measure-preserving transformation of a Lebesgue space is, when restricted to a suitable invariant set of full measure, invertible. By a suitable version of the Jewitt-Krieger Theorem, every zero-entropy ergodic transformation of a Lebesgue space can be represented via measurable isomorphism as a minimal subsystem of the two-sided 2-shift and hence (by the aforementioned corollary) the one-sided 2-shift. By binary coding it follows in particular that every zero-entropy transformation of a Lebesgue space has a representation as an invertible $\times 2$-invariant measure for the doubling map.