Fitting ideals for finitely presented algebraic dynamical systems

Abstract

We consider a class of algebraic dynamical systems introduced by Kitchens and Schmidt. Under a weak finiteness condition -- the Descending Chain Condition -- the dual modules have finite presentations. Using methods from commutative algebra we show how the dynamical properties of the system may be deduced from the Fitting ideals of a finite free resolution of the finitely presented module. The entropy and expansiveness are shown to depend only on the first Fitting ideal (and certain multiplicity data) which gives an easy computation: in particular, no syzygy modules need to be computed. For `square' presentations (in which the number of generators is equal to the number of relations) all the dynamics is visible in the first Fitting ideal and certain multiplicity data, and we show how the dynamical properties and periodic point behaviour may be deduced from the determinant of the matrix of relations.