When $T$ is an irrational rotation, $[T, Id]$ and $[T,T ^{−1}]$ are Bernoulli: explicit isomorphisms

Christophe Leuridan (1, 2, 3)

1 UGA - Université Grenoble Alpes
2 CP - CNRS-périodiques
3 IF - Institut Fourier

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 $[T,T^{ −1}]$.

Mots clés: irrational rotations; skew products; Bernoulli shifts; dyadic filtrations; head-and-tail filtrations; constructive Markov chains; coupling from the past

Domaines: Mathématiques [math]

Christophe Leuridan. When $T$ is an irrational rotation, $[T, Id]$ and $[T,T ^{−1}]$ are Bernoulli: explicit isomorphisms. 2023. ⟨hal-03933493v1⟩