We prove a generalisation of the entropy formula for certain algebraic -actions given in [2] and [4]. This formula expresses the entropy as the logarithm of the Mahler measure of a Laurent polynomial in d variables with integral coefficients. We replace the rational integers by the integers in a number field and examine the entropy of the corresponding dynamical system.
@article{bwmeta1.element.bwnjournal-article-aav88i1p15bwm,
author = {Manfred Einsiedler},
title = {A generalisation of Mahler measure and its application in algebraic dynamical systems},
journal = {Acta Arithmetica},
volume = {89},
year = {1999},
pages = {15-29},
zbl = {0931.11043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav88i1p15bwm}
}
Manfred Einsiedler. A generalisation of Mahler measure and its application in algebraic dynamical systems. Acta Arithmetica, Tome 89 (1999) pp. 15-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav88i1p15bwm/
[000] [1] D. Hilbert, Gesammelte Abhandlungen, Bd. I, Springer, Berlin, 1932.
[001] [2] D. A. Lind, K. Schmidt and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), 593-629. | Zbl 0774.22002
[002] [3] J. Neukirch, Algebraische Zahlentheorie, Springer, Berlin, 1992.
[003] [4] K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser, 1995. | Zbl 0833.28001