Let $\theta $ be an irrational real number. The map $T_\theta : y \mapsto (y+\theta ) \,\mod \!\!\: 1$ from the unit interval $\mathbf {I} = [0,1[$ (endowed with the Lebesgue measure) to itself is ergodic. In 2002, Rudolph and Hoffman showed in [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2) 156(1) (2002), 79–101] that the measure-preserving map $[T_\theta ,\mathrm {Id}]$ is isomorphic to a one-sided dyadic Bernoulli shift. Their proof is not constructive. A few years before, Parry [Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] had provided an explicit isomorphism under the assumption that $\theta $ is extremely well approached by the rational numbers, namely,

$$ \begin{align*}\inf_{q \ge 1} q^44^{q^2}~\mathrm{dist}(\theta,q^{-1}\mathbb{Z}) = 0.\end{align*} $$

Whether the explicit map considered by Parry is an isomorphism or not in the general case was still an open question. In Leuridan [Bernoulliness of $[T,\mathrm {Id}]$ when T is an irrational rotation: towards an explicit isomorphism. Ergod. Th. & Dynam. Sys. 41(7) (2021), 2110–2135] we relaxed Parry’s condition into

$$ \begin{align*}\inf_{q \ge 1} q^4~\mathrm{dist}(\theta,q^{-1}\mathbb{Z}) = 0.\end{align*} $$

In the present paper, we remove the condition by showing that the explicit map considered by Parry is always an isomorphism. With a few adaptations, the same method works with $[T,T^{-1}]$.

Let Tv be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv are obtained. In particular, if the Markov chain is irreducible, then Tv has an rth root when and only when (r, v) = 1.