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Fitting ideals for finitely presented algebraic dynamical systems. (English) Zbl 0972.22005

Let \(G\) be a group acting by automorphisms on a compact abelian group \(X\). The Pontryagin dual \(M=\widehat X\), a discrete abelian group, becomes a module of the group ring \(R=\mathbb{Z}[G]\) in a natural way. Algebraic properties of \(M\) then reflect dynamical and topological properties of the action of \(G\) on \(X\). In the paper under consideration the case \(G=\mathbb{Z}^d\) and \(M\) noetherian is studied. The authors show how entropy, expansiveness and periodic point behaviour depend on the structure of the Fitting ideals of a finite free resolution of \(M\).

MSC:

22D40 Ergodic theory on groups
37A15 General groups of measure-preserving transformations and dynamical systems
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