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 -