As we have seen in Theorem 5.7, every ℤ d -action by automorphisms of a compact, abelian group satisfying the d.c.c. has a dense set of periodic points (Definition 5.5), and Example 5.6 (1) shows that the d.c.c. cannot be dropped in general. In this section we investigate the density of the set of periodic points for a ℤ d -action α by automorphisms of compact group X satisfying the d.c.c. We begin with two examples which show that—if X is non-abelian—the d.c.c. does not necessarily imply that the set of α-periodic points is dense.
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