N ov 2 00 2 ENTROPY GEOMETRY AND DISJOINTNESS FOR ZERO - DIMENSIONAL ALGEBRAIC ACTIONS

We show that many algebraic actions of higher-rank abelian groups on zero-dimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov–Rokhlin formula for half-space entropies. We discuss some mutual disjointness properties of algebraic actions of higher-rank abelian groups on zero-dimensional groups. The tools used are a version of the half-space entropies introduced by Kitchens and Schmidt [14] and adapted by Einsiedler [7], a basic geometric en-tropy formula from [7], and the structure of expansive subdynamics for algebraic Z d-actions due to Einsiedler, Lind, Miles and Ward [9]. We show that any collection of algebraic Z d-actions on zero-dimensional groups with entropy rank or co-rank one that look sufficiently different are mutually disjoint. The main results are the following (here N(·) denotes the set of non-expansive directions defined in Section 1). then the systems are mutually disjoint. The simplest illustration of Theorem 5.1 is the fact that Ledrappier's Example 2.3 and its mirror image are disjoint. This is shown directly in Section 3 to illustrate how the Abramov–Rokhlin formula for half-space entropies may be used. Theorem 6.2. Let Y and Z be prime Z d-actions with entropy co-rank one. If N(α Y) = N(α Z), then Y and Z are disjoint. Once again the simplest illustration of the meaning of this result comes from an example of Ledrappier type: Example 6.3 is a three-dimensional analogue of Ledrappier's example. This is a Z 3-action defined by a 'four-dot' condition which has positive entropy Z 2-subactions; it and its mirror image are disjoint. Surprisingly, it is not the familiar presence of different non-mixing sets but the entropy and subdynamical geometry of the systems that forces this high level of measurable difference of structure. The methods should extend to entropy rank or co-rank greater than one, but the notational and technical difficulties become more substantial. Related work for Z d-actions by toral automorphisms has been done by Kalinin and Katok [11], where more refined information is found about joinings and the consequences of the presence of non-trivial joinings. Actions by toral automorphisms automatically have entropy rank not exceeding one. Our purpose here is to begin to address some of the problems inherent in understanding the joinings between algebraic Z d-actions. The ultimate goal is to extend results like those of [11] to general algebraic actions, just as the rigidity results have been extended …