Ergodic Theory and Dynamical Systems



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


MANFRED EINSIEDLER a1, DOUGLAS LIND a2, RICHARD MILES a3 and THOMAS WARD a3
a1 Mathematisches Institut, Universität Wien, Strudlhofgasse 4, A-1090, Vienna, Austria (e-mail: manfred@mat.univie.ac.at)
a2 Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195-4350, USA (e-mail: lind@math.washington.edu)
a3 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.

(Received June 14 2000)
(Revised September 27 2000)