Duke Mathematical Journal

Topological self-joinings of Cartan actions by toral automorphisms

Elon Lindenstrauss and Zhiren Wang

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Abstract

We show that if r3 and α is a faithful Zr-Cartan action on a torus Td by automorphisms, then any closed subset of (Td)2 which is invariant and topologically transitive under the diagonal Zr-action by α is homogeneous, in the sense that it is either the full torus (Td)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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1336142077

Digital Object Identifier
doi:10.1215/00127094-1593290

Mathematical Reviews number (MathSciNet)
MR2922376

Zentralblatt MATH identifier
1258.37031

Subjects
Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]
Secondary: 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]

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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