This paper is concerned with the entropy of an action of a countable torsion-free abelian group G by continuous automorphisms of a compact abelian group X. A formula is obtained that expresses the entropy in terms of the Mahler measure of a greatest common divisor, complementing earlier work by Einsiedler, Lind, Schmidt and Ward. This leads to a uniform method for calculating entropy whenever G is free. In cases where these methods do not apply, a possible entropy formula is conjectured. The entropy of subactions is examined and, using a theorem of P. Samuel, it is shown that a mixing action of an infinitely generated group of finite rational rank cannot have a finitely generated subaction with finite non-zero entropy. Applications to the concept of entropy rank are also considered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm201-3-4,
author = {Richard Miles},
title = {The entropy of algebraic actions of countable torsion-free abelian groups},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {261-282},
zbl = {1154.37006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm201-3-4}
}
Richard Miles. The entropy of algebraic actions of countable torsion-free abelian groups. Fundamenta Mathematicae, Tome 201 (2008) pp. 261-282. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm201-3-4/