A note on the dynamical zeta function of general toral endomorphisms

It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of g...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2010-08, Vol.161 (1), p.33-42
Main Authors: Baake, Michael, Lau, Eike, Paskunas, Vytautas
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated by integer arithmetic only. We also address the functional equation and the relation between the Artin-Mazur and Lefschetz zeta functions.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-009-0118-y