Entropy range problems and actions of locally normal groups
Let \(G\) be a countable infinite discrete amenable group. The paper considers algebraic \(G\)-actions, i.e.\ continuous automorphisms, preserving Haar-measure \(\mu_X\) of some compact metrizable abelian group \(X\) and studies the metric entropy with respect to \(\mu_X\) as well as the topological entropy of the \(G\)-action. One major task in this context is to determine the set of possible entropies in \([0,\infty]\) for a given family of discrete amenable groups like \({\mathcal G}=\{{\mathbb Z}^d\mid d\geq 1\}\) -- a difficult question related to Lehmer's problem. Denseness in particular corresponds to the existence of actions with arbitrarily small but positive entropy values. To get an answer, the authors establish an entropy formula for locally normal groups of continuous automorphisms acting on finite-dimensional compact abelian groups using a version of Abramov-Rokhlin formula to reduce the setting to the zero-dimensional case as well as Yuzvinskii's addition formula to further reduce to cyclic modules over the group ring \({\mathbb Z}G\). As applications of this formula to entropy range problems they show that for \(G\) containing a non-trivial finite normal subgroup there are algebraic \(G^d\)-actions with arbitrarily small positive entropy; hence the entropy range is dense in \([0,\infty]\) for corresponding families \(\{G^d\mid d\geq 1\}\). Moreover if \(G\) contains arbitrarily large finite normal subgroups the entropy range for \(\{G\}\) already equals \([0,\infty]\). A sufficient condition for \(G\)-actions to have zero entropy is given and as a corollary the authors obtain a special case -- for \(G\) being locally normal -- of the classical result that algebraic \(G\)-actions of completely positive entropy are mixing. The paper ends stating another entropy range problem not covered by the above mentioned results.
Haar-measure preserving algebraic action
scientific article; zbMATH DE number 5629366