Einsiedler, Manfred; Ward, Thomas Entropy geometry and disjointness for zero-dimensional algebraic actions. (English) Zbl 1198.37010 J. Reine Angew. Math. 584, 195-214 (2005). From the text: 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 B. Kitchens and K. Schmidt [Ergodic Theory Dyn. Syst. 9, 691–735 (1989; Zbl 0709.54023)] and adapted by M. Einsiedler [Monatsh. Math. 144, No. 1, 39-69 (2005; Zbl 1061.37006)], a basic geometric entropy formula from that last cited paper, and the structure of expansive subdynamics for algebraic \(\mathbb Z^d\)-actions due to M. Einsiedler, D. Lind, R. Miles and T. Ward [Ergodic Theory Dyn. Syst. 21, 1695–1729 (2001; Zbl 1003.37003)]. We show that any collection of algebraic \(\mathbb Z^d\)-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 \(N(\cdot)\) denotes the set of non-expansive directions defined in the paper).Theorem 5.1. Let \(X_1, \dots, X_n\) be a collection of irreducible algebraic zero-dimensional \(\mathbb Z^d\)-actions, all with entropy rank one. If\[ N(\alpha_j)\backslash \cup_{k>j} N(\alpha_k) \neq\emptyset\;\text{for}\;j = 1, \dots, n \] 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 \(\mathbb Z^d\)-actions with entropy co-rank one. If \(N(\alpha_Y ) \neq N(\alpha_Z)\), 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 \(\mathbb Z^3\)-action defined by a ‘four-dot’ condition which has positive entropy \(\mathbb Z^2\)-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. Cited in 1 ReviewCited in 7 Documents MSC: 37A15 General groups of measure-preserving transformations and dynamical systems 28D20 Entropy and other invariants 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory Keywords:mutual disjointness properties of algebraic actions of higher-rank abelian groups on zero-dimensional groups; Abramov–Rokhlin formula Citations:Zbl 0709.54023; Zbl 1061.37006; Zbl 1003.37003 PDFBibTeX XMLCite \textit{M. Einsiedler} and \textit{T. Ward}, J. Reine Angew. Math. 584, 195--214 (2005; Zbl 1198.37010) Full Text: DOI arXiv Link