This is a beautifully written account of an area of mathematics that has seen rapid development and wide application in the period since Furstenberg introduced the concept of joinings into ergodic theory [H. Furstenberg, Math. Syst. Theory 1, 1–49 (1967; Zbl 0146.28502)]. Indeed the relatively recent dramatic progress perhaps reflects the fact that some of the ideas in Furstenberg’s work were so far ahead of their time that it took some decades for them to be fully appreciated – for example, Katok’s elegant survey of the first half-century of entropy theory [A. Katok, J. Mod. Dyn. 1, No. 4, 545–596 (2007; Zbl 1149.37001)] says “It is interesting to point out that at that time the author (and probably everyone at Moscow) grossly underestimated, almost ignored, Furstenberg’s pioneering disjointness paper” as an intriguing footnote. The objects discussed here are “joinings”: Given a collection of measure-preserving actions of a group \(G\) on Borel probability spaces \((X_i,\mathcal{B}_i,\mu_i)\) for \(i=1,\dots,r\) is a probability measure \(\mu\) on the product space \((X_1\times\cdots\times X_r,\mathcal{B}_1\times\cdots\times\mathcal{B}_r)\) that is invariant under the diagonal action \(g\cdot(x_1,\dots,x_r)=(g\cdot x_1,\dots,g\cdot x_r)\) with the property that the projection of \(\mu\) onto the \(i\)th coordinate is \(\mu_i\) for \(i=1,\dots,r\). The product measure \(\mu_1\times\cdots\times \mu_r\) is a joining and the extent to which other joinings exist is a subtle expression of the way in which the systems are related.
After a brief introduction and history, this survey discusses the work of (mainly) Einsiedler and Lindenstrauss on joinings of higher-rank torus actions on \(S\)-arithmetic homogeneous spaces. One of the broad themes concerns results showing that such systems are either disjoint or strongly related in an algebraic way – measure rigidity. A cogent overview of the methods involved is provided, structured around four main themes: Leafwise measures, Lyapunov exponents, entropy contributions and their connection to invariance; the product structure for coarse Lyapunov weights and the Abramov-Rokhlin formula for coarse weights; the “high entropy” method; and the “low entropy” method. In order to make sense of all this the survey contains an overview of entropy theory in this kind of setting and some of the entropic tools developed by M. Einsiedler et al. [Ann. Math. (2) 164, No. 2, 513–560 (2006; Zbl 1109.22004)] for these problems in Section 2. Corollaries relevant to the structure of joinings are drawn out in Section 3. The high entropy method is discussed in Section 4, with an explanation at a high level of how the material described already gives rise to some profound results. In Section 5, an overview of the low entropy method is given, particularly useful as its formulation “may seem intimidating” in the author’s words. In general, proofs are sketched via motivating examples and there are careful explanations of the ideas as well as the intricate machinery.
The second part of this survey describes some of the many ways in which these joining classification results may be applied. A new application to groups with high rank is also discussed.
This is an attractive survey of some attractive mathematics and will be useful for anyone interested in understanding how the early ideas of Furstenberg have led to such significant mathematics.
For the entire collection see [Zbl 1522.00191].