## Duke Mathematical Journal

### Topological self-joinings of Cartan actions by toral automorphisms

#### Abstract

We show that if $r\geq3$ and $\alpha$ is a faithful $\mathbb {Z}^{r}$-Cartan action on a torus $\mathbb {T}^{d}$ by automorphisms, then any closed subset of $(\mathbb {T}^{d})^{2}$ which is invariant and topologically transitive under the diagonal $\mathbb {Z}^{r}$-action by $\alpha$ is homogeneous, in the sense that it is either the full torus $(\mathbb {T}^{d})^{2}$, or a finite set of rational points, or a finite disjoint union of parallel translates of some $d$-dimensional invariant subtorus. A counterexample is constructed for the rank $2$ case.

#### Article information

Source
Duke Math. J., Volume 161, Number 7 (2012), 1305-1350.

Dates
First available in Project Euclid: 4 May 2012

https://projecteuclid.org/euclid.dmj/1336142077

Digital Object Identifier
doi:10.1215/00127094-1593290

Mathematical Reviews number (MathSciNet)
MR2922376

Zentralblatt MATH identifier
1258.37031

#### Citation

Lindenstrauss, Elon; Wang, Zhiren. Topological self-joinings of Cartan actions by toral automorphisms. Duke Math. J. 161 (2012), no. 7, 1305--1350. doi:10.1215/00127094-1593290. https://projecteuclid.org/euclid.dmj/1336142077

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