The Adjoint Action of an Expansive Algebraic Z–Action

Let d ≥ 1, and let α be an expansive Z-action by continuous automorphisms of a compact abelian group X with completely positive entropy. Then the group ∆α(X) of homoclinic points of α is countable and dense in X, and the restriction of α to the α-invariant subgroup ∆α(X) is a Z-action by automorphisms of ∆α(X). By duality, there exists a Z-action α∗ by automorphisms of the compact abelian group X∗ = ∆̂α(X): this action is called the adjoint action of α. We prove that α∗ is again expansive and has completely positive entropy, and that α and α∗ are weakly algebraically equivalent, i.e. algebraic factors of each other. A Z-action α by automorphisms of a compact abelian group X is reflexive if the Z-action α∗∗ = (α∗)∗ on the compact abelian group X∗∗ = ̂ ∆α∗(X∗) adjoint to α∗ is algebraically conjugate to α. We give an example of a non-reflexive expansive Z-action α with completely positive entropy, but prove that the third adjoint α∗∗∗ = (α∗∗)∗ is always algebraically conjugate to α∗. Furthermore, every expansive and ergodic Z-action α is reflexive. The last section contains a brief discussion of adjoints of certain expansive algebraic Z-actions with zero entropy.