Automorphisms of Compact Abelian Groups and Affine Varieties

This paper investigates the correspondence M↔( X M , α M ), described in [ 6 ], between finitely generated modules M over the ring of Laurent polynomials in d variables with integral coefficients, and pairs ( X , α), where X is a compact, abelian, metrizable group and α: n →α n an is an action of Z d on X such that α n is a continuous automorphism of X for every n ɛ Z d and X satisfies the descending chain condition on closed, α‐invariant subgroups (the descending chain condition is, in particular, satisfied if a is expansive). We characterize ergodicity, expansiveness, and tiniteness of the number of periodic points with any given period of ( X M , α M ) in terms of properties of the module M .