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

Richard Miles. "The entropy of algebraic actions of countable torsion-free abelian groups." Fundamenta Mathematicae 201.3 (2008): 261-282. <http://eudml.org/doc/286257>.

@article{RichardMiles2008,
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.},
author = {Richard Miles},
journal = {Fundamenta Mathematicae},
keywords = {entropy; algebraic action; torsion-free abelian group; Mahler measure; subaction; entropy rank},
language = {eng},
number = {3},
pages = {261-282},
title = {The entropy of algebraic actions of countable torsion-free abelian groups},
url = {http://eudml.org/doc/286257},
volume = {201},
year = {2008},
}

TY - JOUR
AU - Richard Miles
TI - The entropy of algebraic actions of countable torsion-free abelian groups
JO - Fundamenta Mathematicae
PY - 2008
VL - 201
IS - 3
SP - 261
EP - 282
AB - 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.
LA - eng
KW - entropy; algebraic action; torsion-free abelian group; Mahler measure; subaction; entropy rank
UR - http://eudml.org/doc/286257
ER -