A note on the dynamical zeta function of general toral endomorphisms

Abstract

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.

Spectral and topological properties of a family of generalised Thue-Morse sequences

“…Counting finite periodic orbits under the multiplication action on the solenoid, however, means that the inverse limit is not needed, so that the corresponding dynamical (or Artin-Mazur) zeta function coincides with that of the toral endomorphism represented by multiplication by m = k + ℓ ≥ 2. This, in turn, is given by ζ sol m (z) = 1 − z 1 − mz by an application of [11,Thm. 1].…”

“…A natural question concerns the robustness of the singular continuous spectrum under simultaneous permutations of positions in ̺(1) and ̺(1), as considered in [55]. For k = ℓ = 1, the two possible rules are the Thue-Morse rule (11,11) and its partner (11,11), written in obvious shorthand notation. Since the squares of these two rules are equal, they define the same hull, and hence the same autocorrelation.…”

Dynamical invariants for group automorphisms

“…This is discussed in Smale [56,Prop. 4.15], and an elementary algorithmic way to compute the zeta function in integer arithmetic is given by Baake, Lau and Paskunas [5].…”

Recent Trends in Ergodic Theory and Dynamical Systems

“…Let a = a 1 = a 2 , 5) where the last equality follows by the Artin product formula, as n − 1| p = 1. Therefore, whenever n is coprime to…”

“…This is discussed in Smale [56,Prop. 4.15], and an elementary algorithmic way to compute the zeta function in integer arithmetic is given by Baake, Lau and Paskunas [5].…”