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
References:
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References:
 [1] V. I. Arnol'd, Topological and ergodic properties of closed 1-forms with incommensurable periods, Functional Analysis and its Applications, 25 (1991), 81-90.  doi: 10.1007/BF01079587.  Google Scholar [2] G. Forni and A. Kanigowski, Mutliple mixing and disjointness for time changes of bounded-type Heisenberg nilflows, submitted. Google Scholar [3] K. Frączek and M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth functions, Ergodic Theory and Dynam. Systems, 26 (2006), 719-738.  doi: 10.1017/S0143385706000046.  Google Scholar [4] K. Frączek, M. Lemańczyk and E. Lesigne, Mild mixing property for special flows under piecewise constant functions, Discrete Contin. Dyn. Syst., 19 (2007), 691-710.  doi: 10.3934/dcds.2007.19.691.  Google Scholar [5] K. Frączek and M. Lemańczyk, Smooth singular flows in dimension 2 with the minimal self-joining property, Monatsh. Math., 156 (2009), 11-45.  doi: 10.1007/s00605-008-0564-y.  Google Scholar [6] A. Kanigowski, M. Lemańczyk and C. Ulcigrai, On disjointness properties of parabolic flows, Invent. Math. (2020). doi: 10.1007/s00222-019-00940-y.  Google Scholar [7] A. Kanigowski and A. Solomko, On isomorphism problem for von Neumann flows with one discontinuity, Israel Journal of Mathematics, 226 (2018), 685-702.  doi: 10.1007/s11856-018-1701-5.  Google Scholar [8] A. Kanigowski and A. Solomko, On rank of von Neumann special flows, Ergodic Theory and Dynam. Systems, 38 (2018), 2245-2256.  doi: 10.1017/etds.2016.131.  Google Scholar [9] A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310.  doi: 10.1007/BF02760655.  Google Scholar [10] A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook in Dynamical Systems, 1 (2005), 649-743.  doi: 10.1016/S1874-575X(06)80036-6.  Google Scholar [11] A. Kolmogorov, On dynamical systems with an integral invariant on the torus., Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766.   Google Scholar [12] M. Ratner, Rigidity of horocycle flows, Ann. of Math., 115 (1982), 597-614.  doi: 10.2307/2007014.  Google Scholar [13] M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math., 118 (1983), 277-313.  doi: 10.2307/2007030.  Google Scholar [14] M. Ratner, Rigidity of time changes for horocycle flows, Acta Mathematica, 156 (1986), 1-32.  doi: 10.1007/BF02399199.  Google Scholar [15] M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374.  doi: 10.1007/BF01388912.  Google Scholar [16] J. von Neumann, Zur Operatorenmethode in der Klassischen Mechanik, Ann. of Math., 33 (1932), 587-642.  doi: 10.2307/1968537.  Google Scholar
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