A note on the dynamical zeta function of general toral endomorphisms

Abstract

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.

8 pages; revised and slightly expanded version

References in corpus (1)

• Periodic orbits of linear endomorphisms on the 2-torus and its lattices

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• Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices