Fitting ideals for finitely presented algebraic dynamical systems

M. Einsiedler, T. Ward

    Research output: Contribution to journalArticlepeer-review

    7 Citations (Scopus)

    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 resentations. 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 initial 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 initial 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.
    Original languageEnglish
    Pages (from-to)57-71
    Number of pages15
    JournalAequationes Mathematicae
    Volume60
    Issue number1-2
    DOIs
    Publication statusPublished - 2000

    Cite this