Topological self-joinings of cartan actions by toral automorphisms

Elon Lindenstrauss, Zhiren Wang

Research output: Contribution to journalArticlepeer-review


We show that if r ≥ 3 and α is a faithful ℤ( r-Cartan action on a torus T{double-struck} d by automorphisms, then any closed subset of (T{double-struck} d) 2 which is invariant and topologically transitive under the diagonal ℤ r -action by α is homogeneous, in the sense that it is either the full torus (T{double-struck} 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.

Original languageAmerican English
Pages (from-to)1305-1350
Number of pages46
JournalDuke Mathematical Journal
Issue number7
StatePublished - 2012

All Science Journal Classification (ASJC) codes

  • General Mathematics


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