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A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli
Journal d'Analyse Mathématique ( IF 1 ) Pub Date : 2020-08-08 , DOI: 10.1007/s11854-020-0108-5
Marlies Gerber , Philipp Kunde

Let $M$ be a smooth compact connected manifold of dimension $d\geq 2$, possibly with boundary, that admits a smooth effective $\mathbb{T}^2$-action $\mathcal{S}=\left\{S_{\alpha,\beta}\right\}_{(\alpha,\beta) \in \mathbb{T}^2}$ preserving a smooth volume $\nu$, and let $\mathcal{B}$ be the $C^{\infty}$ closure of $\left\{h \circ S_{\alpha,\beta} \circ h^{-1} \;:\;h \in \text{Diff}^{\infty}\left(M,\nu\right), (\alpha,\beta) \in \mathbb{T}^2\right\}$. We construct a $C^{\infty}$ diffeomorphism $T \in \mathcal{B}$ with topological entropy $0$ such that $T \times T$ is loosely Bernoulli. Moreover, we show that the set of such $T \in \mathcal{B}$ contains a dense $G_{\delta}$ subset of $\mathcal{B}$. The proofs are based on a two-dimensional version of the approximation-by-conjugation method.

中文翻译:

一个平滑的零熵微分同胚,其与自身的乘积是松散伯努利

令 $M$ 是维数为 $d\geq 2$ 的光滑紧致连通流形,可能有边界,它承认光滑有效 $\mathbb{T}^2$-action $\mathcal{S}=\left\{ S_{\alpha,\beta}\right\}_{(\alpha,\beta) \in \mathbb{T}^2}$ 保持平滑体积 $\nu$,让 $\mathcal{B}$是 $\left\{h \circ S_{\alpha,\beta} \circ h^{-1} \;:\;h \in \text{Diff}^ 的 $C^{\infty}$ 闭包{\infty}\left(M,\nu\right), (\alpha,\beta) \in \mathbb{T}^2\right\}$。我们构造了一个具有拓扑熵 $0$ 的 $C^{\infty}$ 微分同胚 $T \in \mathcal{B}$,使得 $T \times T$ 是松散的伯努利。此外,我们表明这样的 $T \in \mathcal{B}$ 的集合包含 $\mathcal{B}$ 的密集 $G_{\delta}$ 子集。证明基于共轭逼近方法的二维版本。
更新日期:2020-08-08
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