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

Richard Miles¹

¹ Department of Mathematics KTH SE-100 44 Stockholm, Sweden

Fundamenta Mathematicae, Tome 201 (2008) no. 3, pp. 261-282

Résumé

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.

Détail

DOI : 10.4064/fm201-3-4

Keywords: paper concerned entropy action countable torsion free abelian group continuous automorphisms compact abelian group nbsp formula obtained expresses entropy terms mahler measure greatest common divisor complementing earlier work einsiedler lind schmidt ward leads uniform method calculating entropy whenever cases where these methods apply possible entropy formula conjectured entropy subactions examined using theorem samuel shown mixing action infinitely generated group finite rational rank cannot have finitely generated subaction finite non zero entropy applications concept entropy rank considered