Fuglede–Kadison determinants and entropy for actions of discrete amenable groups

Abstract

In 1990, Lind, Schmidt, and Ward gave a formula for the entropy of certain Z n \mathbb {Z}^n -dynamical systems attached to Laurent polynomials P P , in terms of the (logarithmic) Mahler measure of P P . We extend the expansive case of their result to the noncommutative setting where Z n \mathbb {Z}^n gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede–Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.