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

MILES, Richard (2008). The entropy of algebraic actions of countable torsion-free abelian groups. Fundamenta Mathematicae, 201, 261-282.

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Link to published version:: 10.4064/fm201-3-4

Abstract

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.

Item Type: Article
Departments: Development and Society > Teacher Education
Identification Number: 10.4064/fm201-3-4
Depositing User: Richard Miles
Date Deposited: 26 Jan 2018 14:14
Last Modified: 26 Jan 2018 14:14
URI: http://shura.shu.ac.uk/id/eprint/17232

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