When $T$ is an irrational rotation, $[T, Id]$ and $[T,T ^{−1}]$ are Bernoulli: explicit isomorphisms
Résumé
Let $\theta$ be an irrational real number. The map $T_\theta : y → (y + \theta) \mod 1$ from the unit interval $I = [0, 1[$ (endowed with the Lebesgue measure) to itself is ergodic. In 2002, Rudolph and Hoffman showed in [7] 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 [12] had provided an explicit isomorphism under the assumption that $\theta$ is extremely well approached by the rational numbers, namely $\inf\{q \ge 1 : q^4 4^{q^2} \mathrm{dist}(\theta), q^{−1}\mathbb{Z}) = 0$. Whether the explicit map considered by Parry is an isomorphism or not in the general case was still an open question. In [10] we relaxed Parry's condition into inf $\inf\{q \ge 1 : q^4 \mathrm{dist}(\theta), q^{−1}\mathbb{Z}) = 0$. 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 $[T,T^{ −1}]$.
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