Irreducibility, Homoclinic Points and Adjoint Actions of Algebraic ℤ^d-Actions of Rank One

Abstract

In this paper we consider ℤ^d-actions, d ≥ 1, by automorphisms of compact connected abelian groups which contain at least one expansive automorphism (such actions are called algebraic ℤ^d-actions of expansive rank one). If α is such a ℤ^d-action on an infinite compact connected abelian group X, then every expansive element an of this action has a dense group ${\Delta _{{a^n}}}\left( X \right)$ of homoclinic points. For different expansive elements α^m, αⁿ these groups are generally different and may have zero intersection. By obtaining an appropriate structure formula we prove that these groups are canonically isomorphic for different m, n, and that the restriction of a to any of these groups defines by duality another algebraic ℤ^d-action α* of expansive rank one on a compact connected abelian group X*, called the adjoint action of α. The second adjoint α** = (α*)*obtained by repeating this construction is algebraically conjugate to α.

A class of examples of algebraic ℤ^d-actions of expansive rank one is obtained by fixing a d-tuple c of algebraic numbers consisting not entirely of roots of unity and by associating with it a finite set S_c of places of the algebraic number field K = K(c) generated by the entries of c. For every S_c-integral ideal J in K we define an algebraic ℤ^d-actions α expansive rank one on a compact connected abelian group. In this case the adjoint action α* arises from the inverse ideal class J ^-1 of J. If J,J are two S_c-integral ideals, then the algebraic ℤ^d-actions arising from them are algebraically conjugate if and only if the ideals lie in the same ideal class.

Earlier work in this direction can be found in [4], [6] and [11].