# American Institute of Mathematical Sciences

2020, 16: 37-57. doi: 10.3934/jmd.2020002

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

 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

Received  November 05, 2018 Revised  September 12, 2019 Published  February 2020

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

Citation: Changguang Dong, Adam Kanigowski. Rigidity of a class of smooth singular flows on $\mathbb{T}^2$. Journal of Modern Dynamics, 2020, 16: 37-57. doi: 10.3934/jmd.2020002
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