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arXiv · 1805.01501

Kakutani Equivalence of Unipotent Flows

Adam Kanigowski·Kurt Vinhage·Daren Wei
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Abstract

We study Kakutani equivalence in the class of unipotent flows acting on finite volume quotients of semisimple Lie groups. For every such flow we compute the Kakutani invariant of M. Ratner, the value of which being explicitly given by the Jordan block structure of the unipotent element generating the flow. This in particular answers a question of M. Ratner. Moreover it follows that the only standard unipotent flows are given by $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \times \operatorname{id}$ acting on $(SL(2,\mathbb{R}) \times G')/\Gamma'$, where $\Gamma'$ is an irreducible lattice in $SL(2,\mathbb{R}) \times G'$ (with the possibility that $G' = \{e\}$).

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Adam Kanigowski, Kurt Vinhage, Daren Wei. 2018-05-03. Kakutani Equivalence of Unipotent Flows. https://arxiv.org/abs/1805.01501

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