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 ${\omega }_{\mathcal{X}}$ be the 1-form associated to $\mathcal{X}$. We show that if $|{\int }_{E_{{\mathcal{X}}_1}}d{\omega }_{{\mathcal{X}}_1}|\ne |{\int }_{E_{{\mathcal{X}}_2}}d{\omega }_{{\mathcal{X}}_2}|$, 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 $\alpha ={\alpha }_{\mathcal{X}}\in \mathbb{T}$. We show that for a full measure set of $\alpha \in \mathbb{T}$ the class of smooth time changes of $(v_t^{{\mathcal{X}}_{\alpha }})$ 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.