Homology torsion growth and Mahler measure
\textit{B. Kitchens} and \textit{K. Schmidt} [Ergodic Theory Dyn. Syst. 9, No. 4, 691--735 (1989; Zbl 0709.54023)] associated to any (countable) module \(M\) over the ring \(R=\mathbb{Z}[t_1^{\pm1},\dots,t_d^{\pm1}]\) an action of \(\mathbb{Z}^d\) by automorphisms of the compact (metrizable) abelian dual group of \(M\), initiating a series of results relating the algebraic and geometric properties of the module \(M\) to dynamical properties of the associated \(\mathbb{Z}^d\)-action. \textit{D. Lind} et al. [Invent. Math. 101, No. 3, 593--629 (1990; Zbl 0774.22002)] showed that if \(M\) is finitely generated then the joint topological entropy of the action associated to \(M\) is given by the logarithmic Mahler measure of \(\Delta_0(M)\), the \(0\)th Alexander polynomial of \(M\), and (under the additional hypothesis that the action is expansive) showed that the entropy is given by the growth rate of periodic points along large parallelepipeds. A fundamental feature of many algebraic \(\mathbb{Z}^d\)-actions is however that without expansiveness the set of points fixed by a lattice \(L<\mathbb{Z}^d\) will often be infinite, and Schmidt formulated a conjectural relationship between the entropy of the system associated to the torsion part of the module and the growth rate of the number of connected components of the group of \(L\)-periodic points as \(L\) goes to infinity in a suitable sense. This conjectural relationship is established here, and a related result of \textit{D. S. Silver} and \textit{S. G. Williams} [Topology 41, No. 5, 979--991 (2002; Zbl 1024.57007)] on the growth of homology torsion of finite abelian coverings of link complements is generalized.
scientific article; zbMATH DE number 6361418