Algebraic invariants for group actions on the Cantor set (Q2058971)

This is a very condensed survey of a series of results by many authors concerning Cantor systems, i.e., continuous actions of a countable group \(G\) on the Cantor set \(X\). By an algebraic invariant the author means any algebraic structure which either determines some dynamical properties of the action or depends on the dynamics. The group of automorphisms is an example. The \textit{full group} is another example. It is the subgroup \([G]\) of the group of homeomorphisms \(f\) of \(X\) such that for every \(x\in X\) there exists \(g\in G\) such that \(f(x)=T^g(x)\). More than 40 papers are mentioned in the bibliography. For the entire collection see [\textit{D. R. Wood} (ed.) et al., 2019--20 MATRIX annals. Cham: Springer (2021; Zbl 1459.37002)].