Bernoulliness of $[T, \mathrm{Id}]$ when $T$ is an irrational rotation: towards an explicit isomorphism

Abstract : Let $θ$ be an irrational real number. The map $T_θ : y \mapsto (y + θ) \mod 1$ from the unit interval $I = [0, 1[$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [15] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_θ,\mathrm{Id}]$ and the unilateral dyadic Bernoulli shift when $θ$ is extremely well approached by the rational numbers, namely if $$inf_{q≥1} q^4 4^{q^2} \mathrm{dist}(θ, q^{−1} Z) = 0.$$ A few years later, Rudolph and Hoffman showed in [6] that for every irrational number, the measure-preserving map $[T_θ,\mathrm{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry's condition on $θ$ and show that actually, the explicit map provided by Parry's method is an isomorphism between the map $[T_θ,\mathrm{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$inf_{q≥1} q^4 \mathrm{dist}(θ, q^{−1} Z) = 0.$$ We also provide a weaker sufficient condition involving the expansion of $||θ|| := \mathrm{dist}(θ, Z)$ in continued fraction. Set $||θ|| = [0; a_1 , a_2 ,. . .]$ and call $(p_n /q_n)_{n \ge 0}$ the sequence of convergents. Then Parry's map is an isomorphism between the map $[T_θ,\mathrm{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$inf_{n≥1} q_n^3 (a_1 + · · · + a_n) |q_n θ − p_n | < +∞.$$ Whether Parry's map is an isomorphism for every θ or not is still an open question, although we expect a positive answer.
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Christophe Leuridan. Bernoulliness of $[T, \mathrm{Id}]$ when $T$ is an irrational rotation: towards an explicit isomorphism. 2019. ⟨hal-02272414⟩

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