An example of a weakly mixing $${{\,\textrm{BV}\,}}$$ vector field which is not strongly mixing

Abstract We give an example of a weakly mixing $${{\,\textrm{BV}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>BV</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> divergence-free vector field $$b\in L^\infty ([0,1],{{\,\textrm{BV}\,}}(\mathbb {T}^2))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow> <mml:mspace/> <mml:mtext>BV</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> which is not strongly mixing, in the setting introduced in Bianchini and Zizza (Comm Math Phys 402:1953-2009, 2023), Elgindi and Zlatoš (Adv Math 356, 2019). This is an example of a deterministic vector field with a weakly mixing but not strongly mixing behaviour, while previously were known only examples of strongly mixing vector fields. The construction here is based on a work of Chacon (Proceedings of the American Mathematical Society 22:59-562, 1969) who developed a general framework to generate weakly mixing properties. In particular, he constructed a weakly mixing automorphism (a measure-preserving invertible map) which is not strongly mixing on $$([0,1],\mathcal {B}([0,1]),|\cdot |)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>|</mml:mo> <mml:mo>·</mml:mo> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$\mathcal {B}([0,1])$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are the Borel subsets of [0, 1] and $$|\cdot |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>·</mml:mo> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> is the one-dimensional Lebesgue measure. The idea of the proof is to decompose the motion into simple movements of subquares that can be obtained by composition of divergence-free vector fields. Here the example is given in the two-dimensional torus, after a short revision of the one-dimensional setting, but the construction can be extended to any dimension.