A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli

Abstract

Let $M$ be a smooth compact connected manifold of dimension $d\ge 2$, 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.

52 pages, 8 figures