TY - JOUR
T1 - Topological self-joinings of cartan actions by toral automorphisms
AU - Lindenstrauss, Elon
AU - Wang, Zhiren
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84864312294&partnerID=8YFLogxK
U2 - 10.1215/00127094-1593290
DO - 10.1215/00127094-1593290
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AN - SCOPUS:84864312294
SN - 0012-7094
VL - 161
SP - 1305
EP - 1350
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 7
ER -