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Expansive automorphisms of compact groups

  • Klaus Schmidt
Chapter
Part of the Progress in Mathematics book series (PM, volume 128)

Abstract

In the Sections 7–8 we investigated the structure of ℤ d -actions of the form \( {\alpha^{{{\Re_d}/\mathfrak{p}}}} \), where p ⊂ ℜ d is a prime ideal. Although we can find, for every ℤ d -action α by automorphisms of a compact, abelian group X, a sequence of closed, α-invariant subgroups X = Y 0 Y 1 ⊃ ... such that \( {\alpha^{{{Y_j}/{Y_{{j + 1}}}}}} \) is of the form \( {\alpha^{{{Y_j}/{Y_{{j + 1}}}}}} \) for every j ≥ 0, where (p j ) is a sequence of prime ideals in ℜ d (Corollary 6.2), the reconstruction of α from these quotient-actions is a problem of formidable difficulty. Only when d = 1 can one ‘almost’ re-build the action α from the quotient actions \( {\alpha^{{{\Re_d}/{\mathfrak{q}_j}}}} \) (Corollary 9.4), due to the fact that ℚ ⊗1 = ℚu 1 ±1 is a principal ideal domain. The main tool in this reconstruction is the following Lemma 9.1.

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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Klaus Schmidt
    • 1
  1. 1.Mathematisches InstitutUniversität WienViennaAustria

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