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$ {\rm GL}(2,{\bf Z})$ action on a two torus


Author: Kyewon Koh Park
Journal: Proc. Amer. Math. Soc. 114 (1992), 955-963
MSC: Primary 58F11; Secondary 28D15, 58F17
DOI: https://doi.org/10.1090/S0002-9939-1992-1059631-6
MathSciNet review: 1059631
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Abstract: We study the group $ \operatorname{GL}(2,Z)$ action on a two torus $ {T^2}$ with Lebesgue measure. We show that any measure-preserving transformation that commutes with the group action is the action of a matrix of the form $ \left( {\begin{array}{*{20}{c}} m & 0 \\ 0 & m \\ \end{array} } \right)$, where $ m$ is an integer. We also show that all factors come from these commuting transformations. Finally we show that the set of self-joinings consists of the product measure and the measures sitting on the graphs $ (Ku,Mu):K = \left( {\begin{array}{*{20}{c}} k & 0 \\ 0 & k \\ \end{array} } \right),M = \left( {\begin{array}{*{20}{c}} m & 0 \\ 0 & m \\ \end{array} } \right)$, and $ u \in {T^2}\} $. We provide an example whose self-joinings consist only of the product measure and the diagonal measure.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1059631-6
Article copyright: © Copyright 1992 American Mathematical Society

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