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

Published online by Cambridge University Press:  28 November 2001

MANFRED EINSIEDLER
Affiliation:
Mathematisches Institut, Universität Wien, Strudlhofgasse 4, A-1090, Vienna, Austria (e-mail: manfred@mat.univie.ac.at)
DOUGLAS LIND
Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195-4350, USA (e-mail: lind@math.washington.edu)
RICHARD MILES
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK (e-mail: {r.miles,t.ward}@uea.ac.uk)
THOMAS WARD
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK (e-mail: {r.miles,t.ward}@uea.ac.uk)

Abstract

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.

Type
Research Article
Copyright
2001 Cambridge University Press

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