Dynamical systems arising from units in Krull rings (Q5931966)

A class of dynamical systems associated to rings in algebraic number fields and rational function fields with transcendence degree one (the \(S\)-integer systems) were studied by \textit{V. Chothi, G. Everest} and the reviewer [J. Reine Angew. Math. 489, 99-132 (1997; Zbl 0879.58037)] and the reviewer [Ergodic Theory Dyn. Syst. 18, No. 2, 471-486 (1998; Zbl 0915.58081)]. In this interesting paper some of these results are generalised to the higher-rank setting, in which a \({\mathbb Z}^d\)-action is associated to a \(d\)-tuple of units in the subring of a field which may have positive transcendence degree \(\rho\) over the rationals. There are serious technical difficulties in making this extension. The author has used non-trivial properties of Krull rings to show that many dynamical properties can be interpreted in terms of valuations even in the higher-rank setting. That, for example, expansiveness of such actions for \(d>1\) can be characterised in terms of valuations is a quite unexpected result which will be important in understanding other properties of these systems, for example the structure of expansive sub-actions. A major result (Theorem 4.6) is that the entropy of the \({\mathbb Z}^d\)-action must be zero if \(d>\rho\), must be infinite if \(d<\rho\) and for the critical case \(d=\rho\) examples are given to show that the entropy may be zero, infinite, or positive and finite. Reviewer's remark: Additional examples and results may be found in the author's PhD thesis [Univ. East Anglia PhD. thesis (2000); unpublished].