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

TL;DR

This paper constructs smooth, zero-entropy diffeomorphisms on certain manifolds whose product exhibits loosely Bernoulli behavior, revealing new dynamical properties and density results within a specific conjugacy class.

Contribution

It introduces a method to produce smooth zero-entropy diffeomorphisms with loosely Bernoulli products, expanding understanding of complex dynamical systems on manifolds with torus actions.

Findings

Existence of smooth zero-entropy diffeomorphisms with Bernoulli products

Density of such diffeomorphisms in a conjugacy class

Application of a two-dimensional approximation-by-conjugation method

Abstract

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.