Expansive subdynamics for algebraic \(\mathbb{Z}^d\)-actions. (English) Zbl 1003.37003

A general framework for investigating topological actions for \(\mathbb{Z}^n\) actions on compact metric spaces was proposed by M. Boyle and D. Lind [Trans. Am. Math. Soc. 349, No. 1, 55–102 (1997; Zbl 0863.54034)]. Here the notion of expansiveness along lower-dimensional subspaces of \(\mathbb{R}^d\) is the starting point for the study of subactions and their abrupt behaviour changes at the border of the generalized Weyl chambers. In this interesting paper the authors give a complete characterization of expansiveness of subspaces in the case of algebraic actions on compact abelian groups.
Such an action corresponds via Pontryagin duality to a module over the ring of Laurent polynomials with coefficients in \(\mathbb{Z}\). Many dynamical properties have algebraic characterizations using this module [see K. Schmidt’s book ‘Dynamical systems of algebraic origin’, Prog. Math. 128, Birkhäuser (1995; Zbl 0833.28001)].
The description of expansiveness for subspaces uses the logarithmic image of a variety and a directional version of Noetherianess. The first is also known as amoeba and has been studied extensively over the last years, see AMS Notices September 2000. The latter is related to a geometric invariant originally introduced by Bieri and Strebel for group theory, and can be computed using universal Gröbner bases.
The stability of the homoclinic equivalence relation inside the generalized Weyl chambers is proved, and the abrupt change on the boundary is shown in an example. Furthermore, several notions of dynamical rank for algebraic actions are introduced, and the theory is illustrated in many examples.


37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B99 Topological dynamics
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
54H15 Transformation groups and semigroups (topological aspects)
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