Dynamical Systems of Algebraic Origin pp 77-92 | Cite as

# 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} ℜ_{1} = ℚ*u* _{1} ^{±1} is a principal ideal domain. The main tool in this reconstruction is the following Lemma 9.1.

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