Abstract
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 language | American English |
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Pages (from-to) | 1305-1350 |
Number of pages | 46 |
Journal | Duke Mathematical Journal |
Volume | 161 |
Issue number | 7 |
DOIs | |
State | Published - 2012 |