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  • Cited by 8
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    Bieri, Robert and Geoghegan, Ross 2016. Limit sets for modules over groups onCAT(0) spaces: from the Euclidean to the hyperbolic. Proceedings of the London Mathematical Society, Vol. 112, Issue. 6, p. 1059.

    Ward, Thomas and Miles, Richard 2015. Directional uniformities, periodic points, and entropy. Discrete and Continuous Dynamical Systems - Series B, Vol. 20, Issue. 10, p. 3525.

    Theobald, Thorsten and de Wolff, Timo 2013. Amoebas of genus at most one. Advances in Mathematics, Vol. 239, p. 190.

    Meyerovitch, Tom 2011. Growth-type invariants for ℤ d subshifts of finite type and arithmetical classes of real numbers. Inventiones mathematicae, Vol. 184, Issue. 3, p. 567.

    Ward, Thomas and Miles, Richard 2011. A directional uniformity of periodic point distribution and mixing. Discrete and Continuous Dynamical Systems, Vol. 30, Issue. 4, p. 1181.

    Miles, Richard 2006. Expansive Algebraic Actions of Countable Abelian Groups. Monatshefte für Mathematik, Vol. 147, Issue. 2, p. 155.

    Einsiedler, Manfred and Ward, Thomas 2005. Isomorphism rigidity in entropy rank two. Israel Journal of Mathematics, Vol. 147, Issue. 1, p. 269.

    Bhattacharya, Siddhartha and Schmidt, Klaus 2003. Homoclinic points and isomorphism rigidity of algebraic ℤ d -actions on zero-dimensional compact abelian groups. Israel Journal of Mathematics, Vol. 137, Issue. 1, p. 189.

  • Ergodic Theory and Dynamical Systems, Volume 21, Issue 6
  • December 2001, pp. 1695-1729

Expansive subdynamics for algebraic \mathbb{Z}^d-actions

  • DOI:
  • Published online: 28 November 2001

A general framework for investigating topological actions of \mathbb{Z}^d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of \mathbb{R}^d. Here we completely describe this expansive behavior for the class of algebraic \mathbb{Z}^d-actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables.

We introduce two notions of rank for topological \mathbb{Z}^d-actions, and for algebraic \mathbb{Z}^d-actions describe how they are related to each other and to Krull dimension. For a linear subspace of \mathbb{R}^d we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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