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.

中文翻译：

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何时的伯努利是一个非理性的旋转：朝向一个明确的同构

让$\unicode[STIX]{x1D703}$是一个无理数。地图$T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$从单位区间$\mathbf{I}= [\!0,1\![$（赋予勒贝格测度）本身是遍历的。在一篇简短的论文中 [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products。埃尔戈德。钍。&动态。系统。16(1996), 519–529] 发表于 1996 年，Parry 提供了一个显式的同构$[T_{\unicode[STIX]{x1D703}},\text{Id}]$和单边二元伯努利位移，当$\unicode[STIX]{x1D703}$由有理数非常近似，即，如果$$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{- 1}\mathbb{Z})=0.\end{eqnarray}$$几年后，霍夫曼和鲁道夫[与伯努利位移同构的均匀自同态。安。数学。(2)156(2002), 79–101] 表明，对于每个无理数，保测映射$[T_{\unicode[STIX]{x1D703}},\text{Id}]$与单边二元伯努利位移同构。他们的证明不具建设性。在本文中，我们显着放宽了 Parry 的条件$\unicode[STIX]{x1D703}$: Parry 方法提供的显式映射是映射之间的同构$[T_{\unicode[STIX]{x1D703}},\text{Id}]$单边二元伯努利位移$$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})= 0.\end{eqnarray}$$这个条件可以再放宽到$$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX] {x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$在哪里$[0;a_{1},a_{2},\ldots ]$是连分数展开式和$(p_{n}/q_{n})_{n\geq 0}$的收敛序列$\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Parry 的映射是否是每个的同构$\unicode[STIX]{x1D703}$与否仍然是一个悬而未决的问题，尽管我们期待一个肯定的答案。