Expansive automorphisms of compact groups

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 ⊗_ℤ ℜ₁ = ℚu ^±1₁ is a principal ideal domain. The main tool in this reconstruction is the following Lemma 9.1.