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

Author(s): Christopher Deninger
Journal: J. Amer. Math. Soc. 19 (2006), 737-758.
MSC (2000): Primary 22D25, 37A35, 37B40, 46Lxx
Posted: February 6, 2006
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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.

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Additional Information:

Christopher Deninger
Affiliation: Mathematisches Institut, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email: c.deninger@math.uni-muenster.de

DOI: 10.1090/S0894-0347-06-00519-4
PII: S 0894-0347(06)00519-4
Keywords: Fuglede--Kadison determinant, entropy, amenable group, Mahler measure
Received by editor(s): April 11, 2005
Posted: February 6, 2006
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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