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Automorphisms of Compact Abelian Groups and Affine Varieties
Author(s) -
Schmidt Klaus
Publication year - 1990
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-61.3.480
Subject(s) - mathematics , automorphism , abelian group , metrization theorem , pure mathematics , nilpotent , invariant (physics) , finitely generated abelian group , contractible space , center (category theory) , locally compact space , discrete mathematics , mathematical analysis , separable space , mathematical physics , chemistry , crystallography
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 .