The entropy of algebraic actions of countable torsion-free abelian groups

Volume 201 / 2008

Richard Miles Fundamenta Mathematicae 201 (2008), 261-282 MSC: 37A35, 37B40. DOI: 10.4064/fm201-3-4


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.


  • Richard MilesDepartment of Mathematics
    SE-100 44 Stockholm, Sweden

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