Duke Mathematical Journal

Topological self-joinings of Cartan actions by toral automorphisms

Elon Lindenstrauss and Zhiren Wang

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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.

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Duke Math. J., Volume 161, Number 7 (2012), 1305-1350.

First available in Project Euclid: 4 May 2012

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Zentralblatt MATH identifier

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]


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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  • [B] D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), 509–532.
  • [EL1] M. Einsiedler and E. Lindenstrauss, Rigidity properties ofd-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 99–110.
  • [EL2] M. Einsiedler and D. Lind, Algebraicd-actions on entropy rank one, Trans. Amer. Math. Soc. 356 (2004), 1799–1831.
  • [F] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49.
  • [G] E. Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr. 101, Amer. Math. Soc., Providence, 2003.
  • [GH] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 36, Amer. Math. Soc., Providence, 1955.
  • [KK] B. Kalinin and A. Katok, Measurable rigidity and disjointness fork actions by toral automorphisms, Ergodic Theory Dynam. Systems 22 (2002), 507–523.
  • [LS] E. Lindenstrauss and U. Shapira, Homogeneous orbit closures and applications, Ergodic Theory Dynam. Systems 32 (2011), 785–807.
  • [Mar] G. Margulis, “Problems and conjectures in rigidity theory” in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, 2000, 161–174.
  • [Mau] F. Maucourant, A nonhomogeneous orbit closure of a diagonal subgroup, Ann. of Math. (2) 171 (2010), 557–570.
  • [P] C. J. Parry, Units of algebraic numberfields, J. Number Theory 7 (1975), 385–388.
  • [Sch] K. Schmidt, Dynamical Systems of Algebraic Origin, Progr. Math. 128, Birkhäuser, Basel, 1995.
  • [Sha] U. Shapira, A solution to a problem of Cassels and Diophantine properties of cubic numbers, Ann. of Math. (2) 173 (2011), 543–557.
  • [T] G. Tomanov, Locally divergent orbits on Hilbert modular spaces, to appear in Int. Math. Res. Not. IMRN, preprint, arXiv:1012.6006v3 [math.DS]
  • [W1] Z. Wang, Quantitative density under higher rank abelian algebraic toral actions, Int. Math. Res. Not. IMRN 2011, no. 16, 3744–3821.
  • [W2] Z. Wang, Rigidity of commutative non-hyperbolic actions by toral automorphisms, to appear in Ergodic Theory Dynam. Systems, preprint, arXiv:1101.0321v2 [math.DS]