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 ${\mathbb{Z}}^n$-dynamical systems attached to Laurent polynomials $P$, in terms of the (logarithmic) Mahler measure of $P$. We extend the expansive case of their result to the noncommutative setting where ${\mathbb{Z}}^n$ 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.

27 pages

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