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Zero entropy

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

Abstract

One of the most interesting phenomena which arises in the transition from ℤ-actions to ℤ d -actions by automorphisms of compact groups is the existence of non-trivial (e.g. mixing) actions with zero entropy. Theorem 19.5 characterizes the principal prime ideals p ⊂ ℜ d which are null, and from Corollary 18.5 we know that every non-principal prime ideal p ⊂ ℜ d is null. Theorem 6.5 (2) and Proposition 19.4 yield an abundance of mixing ℤ d -actions with zero entropy. One class of such actions on compact, connected, abelian groups is introduced in Section 7, where we investigate actions of the form \({\alpha^{{{\Re_{{{{d} \left/ {a} \right.}}}}}}}\) arising from prime ideals a ⊂ ℜ d for which V(a) is finite, and other zero entropy ℤ d -actions are considered in the Examples 4.16 (1), 5.3 (5), 6.18 (5), and 8.5 (1). Before introducing further examples of mixing ℤ d with zero entropy we shall discuss briefly the restrictions α(Г)of a ℤ d -action α by automorphisms of a compact, abelian group X to various subgroups Г ⊂ ℤ d . If h(α) = 0, then h(α (Г) ) may be positive (even infinite) for certain subgroups Г ⊂ ℤ d of rank r < d. For a ℤ d -action of the form \(\alpha = {\alpha^{{\Re {{d} \left/ {p} \right.}}}}\), where p ⊂ ℜd is a prime ideal, this dependence of entropy on the rank of 0413involves the number r(p) introduced in the Propositions 8.2–8.3.

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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

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

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