Dynamical Systems of Algebraic Origin pp 221-259 | Cite as

# Zero entropy

## 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*_{ℂ}(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 ⊂ ℜ_{d} 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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