The entropy of algebraic actions of countable torsion-free abelian groups
2008 (English)In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 201, no 3, 261-282 p.Article in journal (Refereed) Published
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.
Place, publisher, year, edition, pages
2008. Vol. 201, no 3, 261-282 p.
entropy, algebraic action, torsion-free abelian group, Mahler measure, subaction, entropy rank
IdentifiersURN: urn:nbn:se:kth:diva-36335ISI: 000264414700004ScopusID: 2-s2.0-59549099883OAI: oai:DiVA.org:kth-36335DiVA: diva2:430623
FunderKnut and Alice Wallenberg Foundation, KAW 2005.0098
QC 201107112011-07-112011-07-112011-07-11Bibliographically approved