Homoclinic points of algebraic $\mathbb {Z}^d$-actions

Abstract

An algebraic zd-action is an action of Zd by (continuous) automorphisms of a compact abelian group. The dynamics of a single group automorphism have been investigated in great detail over the past several decades. More recently, the study of algebraic Zd-actions for d > 2 has revealed a striking interplay between these actions and commutative algebra. In ?2 we summarize those parts of this interaction needed here. The purpose of this paper is to study the homoclinic points of algebraic Zd_ actions. Let al be an algebraic Zd-action on the compact abelian group X, and let Ox denote the additive identity of X. A point x E X is homoclinic for ce if c nX -? Ox as llnll -oo. The set Ai(X) of all homoclinic points for Oc is clearly a subgroup of X which we call the homoclinic group of oz. In ?3 we discuss some elementary properties of the homoclinic group, including counltability of ZA (X) whenever oz is expansive. Our two main results are contained in ?4. These are that if oz is an expansive algebraic Ed-action, then (1) A,(X) is nontrivial if and only if oz has (strictly) positive entropy, and (2) A, (X) is nontrivial and dense in X if and only if oz has completely positive entropy. The second result is proved by first establishing in Lemma 4.5 the density of A, (X) for certain expansive actions by use of Fourier analysis. For these actions A, (X) is generated by a single fundamental homoclinic point which can be computed explicitly. This lemma is then combined with some commutative algebra to prove (2). Recent work of Kaminker and Putnam [3], [11] has suggested a general duality in the K-theory of C*-algebras. For a principal expansive action al on X we show that A, (X) is isomorphic to the dual group of X, providing a class of examples to which their duality theory applies. Ruelle has investigated expansive topological Zd-actions which satisfy an orbit tracing property called specification (see [12] and [13]), showing that there is a thermodynamic formalism for such actions. In ?5 we show that expansive algebraic Zd-actions with completely positive entropy always satisfy very strong specification properties, thereby providing an extensive class of examples to which the thermodynamic formalism applies.