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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold V.I., Avez A.: Ergodic Problems of Classical Mechanics. Reprint, Addison-Wesley, Redwood City (1989)
Adler R.L., Palais R.: Homeomorphic conjugacy of automorphisms of the torus. Proc. AMS 16, 1222–1225 (1965)
Artin M., Mazur B.: On periodic points. Ann. Math. 81, 82–99 (1965)
Baake, M., Hermisson, J., Pleasants, P.A.B.: The torus parametrization of quasiperiodic LI-classes. J. Phys. A Math. Gen. 30, 3029–3056 (1997). mp_arc/02-168
Baake M., Roberts J.A.G.: Symmetries and reversing symmetries of toral automorphisms. Nonlinearity 14, R1–R24 (2001) arXiv:math.DS/0006092
Baake M., Roberts J.A.G., Weiss A.: Periodic orbits of linear endomorphisms on the 2-torus and its lattices. Nonlinearity 21, 2427–2446 (2008) arXiv:0808.3489
Bowen, R., Lanford III, O.E.: Zeta functions of restrictions of the shift transformation. In: Global Analysis, vol. 14 of Proc. Sympos. Pure Math., pp. 43–49. AMS, Providence (1970)
Bredon G.: Topology and Geometry. Springer, Berlin (1993)
Brooks R.B.S., Brown R.F., Pak J., Taylor D.H.: Nielsen numbers of maps of tori. Proc. AMS 52, 398–400 (1975)
Chothi V., Everest G., Ward T.: S-integer dynamical systems: periodic points. J. Reine Angew. Math (Crelle) 489, 99–132 (1997)
Degli Esposti M., Isola S.: Distribution of closed orbits for linear automorphisms of tori. Nonlinearity 8, 827–842 (1995)
Dold A.: Lectures on Algebraic Topology. Springer, Berlin (1972)
Fel’shtyn, A.: Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion. Memoirs AMS vol. 147, no. 699, AMS, Providence (2000)
Gantmacher F.: The Theory of Matrices. Chelsea, New York (1959)
Katok A., Hasselblatt B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)
Lehmer D.H.: Factorization of certain cyclotomic functions. Ann. Math (2) 34, 461–479 (1933)
Manning A.: Axiom A diffeomorphisms have rational zeta functions. Bull. Lond. Math. Soc. 3, 215–220 (1971)
Miles R.: Zeta functions for elements of entropy rank one actions. Ergod. Theory Dyn. Syst. 27, 567–582 (2007)
Puri, Y., Ward, T.: Arithmetic and growth of periodic orbits. J. Integer Sequences 4, (2001), paper 01.2.1
Ruelle, D.: Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval. CRM Monograph Series, vol. 4, AMS, Providence (1994)
Ruelle, D.: Dynamical zeta functions and transfer operators. Preprint IHES/M/02/66 (2002)
Smale S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)
Waddington S.: The prime orbit theorem for quasihyperbolic toral automorphisms. Monatsh. Math. 112, 235–248 (1991)
Walters P.: An Introduction to Ergodic Theory. Reprint, Springer, New York (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. G. Dani.
Rights and permissions
About this article
Cite this article
Baake, M., Lau, E. & Paskunas, V. A note on the dynamical zeta function of general toral endomorphisms. Monatsh Math 161, 33–42 (2010). https://doi.org/10.1007/s00605-009-0118-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-009-0118-y