Rigidity of a class of smooth singular flows on \begin{document}$ \mathbb{T}^2 $\end{document}

Abstract

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field \begin{document}$ \mathscr{X} $\end{document} on \begin{document}$ \mathbb{T}^2\setminus \{a\} $\end{document} , where \begin{document}$ \mathscr{X} $\end{document} is not defined at \begin{document}$ a\in \mathbb{T}^2 $\end{document} and \begin{document}$ \mathscr{X} $\end{document} has one critical point which is a center. It follows that the phase space can be decomposed into a (topological disc) \begin{document}$ D_\mathscr{X} $\end{document} and an ergodic component \begin{document}$ E_\mathscr{X} = \mathbb{T}^2\setminus D_\mathscr{X} $\end{document} . Let \begin{document}$ \omega_\mathscr{X} $\end{document} be the 1-form associated to \begin{document}$ \mathscr{X} $\end{document} . We show that if \begin{document}$ |\int_{E_{\mathscr{X}_1}}d\omega_{\mathscr{X}_1}|\neq |\int_{E_{\mathscr{X}_2}}d\omega_{\mathscr{X}_2}| $\end{document} , then the corresponding flows \begin{document}$ (v_t^{\mathscr{X}_1}) $\end{document} and \begin{document}$ (v_t^{\mathscr{X}_2}) $\end{document} are disjoint. It also follows that for every \begin{document}$ \mathscr{X} $\end{document} there is a uniquely associated frequency \begin{document}$ \alpha = \alpha_{\mathscr{X}}\in \mathbb{T} $\end{document} . We show that for a full measure set of \begin{document}$ \alpha\in \mathbb{T} $\end{document} the class of smooth time changes of \begin{document}$ (v_t^\mathscr{X_ \alpha}) $\end{document} 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 [ 15 ,Problem 3] is positive.