Journal Article

Bernoulliness of when is an irrational rotation: towards an explicit isomorphism

LEURIDAN, CHRISTOPHE

2021

Overview

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$$\\begin{eqnarray}ınf _{q\\geq 1}q^{4}4^{q^{2}}~\\text{dist}(\\unicode[STIX]{x1D703},q^{-1}\\mathbb{Z})=0.\\end{eqnarray}$$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$$\\begin{eqnarray}ınf _{q\\geq 1}q^{4}~\\text{dist}(\\unicode[STIX]{x1D703},q^{-1}\\mathbb{Z})=0.\\end{eqnarray}$$This condition can be relaxed again into$$\\begin{eqnarray}ınf _{n\\geq 1}q_{n}^{3}~(a_{1}+\\cdots +a_{n})~|q_{n}\\unicode[STIX]{x1D703}-p_{n}|<+ınfty ,\\end{eqnarray}$$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.

• Publisher: Cambridge University Press
• Subject: Automorphisms / Irrationality / Isomorphism / Numbers