$\textrm {GL}(2,\textbf {Z})$ action on a two torus
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- by Kyewon Koh Park PDF
- Proc. Amer. Math. Soc. 114 (1992), 955-963 Request permission
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.References
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Additional Information
- © Copyright 1992 American Mathematical Society
- 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