Rigidity of a class of smooth singular flows on $\mathbb T^2$

Abstract

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\mathcal{X}$ on $\mathbb T^2\setminus \{a\}$, where $\mathcal{X}$ is not defined at $a\in \mathbb T^2$. It follows that the phase space can be decomposed into a (topological disc) $D_\mathcal{X}$ and an ergodic component $E_\mathcal{X}=\mathbb T^2\setminus D_\mathcal{X}$. Let $ω_\mathcal{X}$ be the 1-form associated to $\mathcal{X}$. We show that if $|\int_{E_{\mathcal{X}_1}}dω_{\mathcal{X}_1}|\neq |\int_{E_{\mathcal{X}_2}}dω_{\mathcal{X}_2}|$, then the corresponding flows $(v_t^{\mathcal{X}_1})$ and $(v_t^{\mathcal{X}_2})$ are disjoint. It also follows that for every $\mathcal{X}$ there is a uniquely associated frequency $α=α_{\mathcal{X}}\in \mathbb T$. We show that for a full measure set of $α\in \mathbb T$ the class of smooth time changes of $(v_t^\mathcal{X_α})$ is joining rigid, i.e. every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to a problem of Ratner (Problem 3 in \cite{Rat4}) is positive.