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.