The adjoint action of an expansive algebraic \(Z^d\)-Action (Q1601796)

Let \(\alpha: \mathbb{Z}^d\to \text{Aut}(X)\), \(\alpha({\mathbf n})= \alpha^{{\mathbf n}}\), be a homomorphism, \(X\) a compact Abelian group. \(\alpha\) is an expansive algebraic \(\mathbb{Z}^d\)-action if there exists open \(O\subset X\) (a neighborhood of the identity \(0_X\in X\)) with \(\bigcap_{{\mathbf n}\in \mathbb{Z}^d} \alpha^{-n}(O)= \{0_X\}\). If \(\alpha\) is a single expansive (and hence ergodic) automorphism of the \(m\)-torus \({\mathbf T}^m\), the adjoint of \(\alpha\) is the automorphism \(\alpha^*\) of \({\mathbf T}^m\) defined by the transpose of the matrix \(A\) defining \(\alpha\). The general notion of adjoint \(\alpha^*\) of an algebraic \(\mathbb{Z}^d\)-action is studied, and is shown to have completely positive entropy (although \(\alpha\) may not). \(\alpha\) and \(\alpha^*\) have the same entropy and \(\alpha\) is conjugate to \(\alpha^{***}\), but need not be conjugate to \(\alpha^{**}\) (reflexive), but those resulting from single automorphisms are always reflexive. Finally, a more general notion of adjoint action is mentioned for arbitrary expansive algebraic \(\mathbb{Z}^d\)-actions with not necessarily positive entropy.