An example of a weakly mixing BV vector field which is not strongly mixing

TL;DR

This paper constructs a specific example of a vector field on the 2D torus that is weakly mixing but not strongly mixing, illustrating a nuanced distinction in dynamical systems.

Contribution

It provides the first explicit example of a BV vector field on the torus that exhibits weak mixing without strong mixing, extending Chacon's automorphism example.

Findings

The constructed vector field is weakly mixing.

The vector field is not strongly mixing.

The example is in the BV function space.

Abstract

We give an example of a weakly mixing vector field $b\in L^\infty([0,1],\text{BV}(\mathbb{T}^2))$ which is not strongly mixing. The example is based on a work of Chacon who constructed a weakly mixing automorphism which is not strongly mixing on $([0,1],\mathcal{B}([0,1]),|\cdot|)$, where $\mathcal{B}([0,1])$ are the Borel subsets of $[0,1]$ and $|\cdot|$ is the one-dimensional Lebesgue measure.