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

Abstract

Let $\\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\\unicode[STIX]{x1D703}}:y\\mapsto (y+\\unicode[STIX]{x1D703})\\!\\hspace{0.6em}{\\rm mod}\\hspace{0.2em}1$ from the unit interval $\\mathbf{I}= [\\!0,1\\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys. 16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\\unicode[STIX]{x1D703}},\\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\\begin{eqnarray}\\inf _{q\\geq 1}q^{4}4^{q^{2}}~\\text{dist}(\\unicode[STIX]{x1D703},q^{-1}\\mathbb{Z})=0.\\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2) 156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\\unicode[STIX]{x1D703}},\\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\\unicode[STIX]{x1D703}$ : the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\\unicode[STIX]{x1D703}},\\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\\begin{eqnarray}\\inf _{q\\geq 1}q^{4}~\\text{dist}(\\unicode[STIX]{x1D703},q^{-1}\\mathbb{Z})=0.\\end{eqnarray}$$ This condition can be relaxed again into $$\\begin{eqnarray}\\inf _{n\\geq 1}q_{n}^{3}~(a_{1}+\\cdots +a_{n})~|q_{n}\\unicode[STIX]{x1D703}-p_{n}|<+\\infty ,\\end{eqnarray}$$ where $[0;a_{1},a_{2},\\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\\geq 0}$ the sequence of convergents of $\\Vert \\unicode[STIX]{x1D703}\\Vert :=\\text{dist}(\\unicode[STIX]{x1D703},\\mathbb{Z})$ . Whether Parry’s map is an isomorphism for every $\\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.