The torus parametrization of quasiperiodic LI-classes

Pure point diffraction and cut and project schemes for measures: the smooth case

“…They have been introduced in [2,3], to which we refer for further details and proofs of the subsequent discussion. A measure ν ∈ M(G) is called translation bounded if there exist some C > 0 and an open non empty relatively compact set V in G so that (1) |ν|(t + V ) ≤ C for every t ∈ G, where |ν| is the total variation measure of ν. The set of all translation bounded measures satisfying (1) is denoted by M C,V (G).…”

A note on the dynamical zeta function of general toral endomorphisms

“…This formula is well-known [24,4] Following [3], the Artin-Mazur zeta function of a general M ∈ Mat(d, Z) is defined as…”

“…Let us briefly illustrate this phenomenon in one dimension. Endomorphisms of T 1 ≃ S 1 are represented by multiplication (mod 1) with an integer n. The dynamical zeta function reads ζ 0 (z) = 1/(1 − z) and (4) ζ n (z) = 1 − sgn(n)z 1 − |n|z for n = 0, due to our Theorem 1 below (or a simple direct calculation). For n = −1, we get ζ −1 (z) = (1 − z 2 )/(1 − z) 2 , thus c 1 = 2 and c 2 = −1, while c m = 0 for all m ≥ 3.…”

A Note on Palindromicity

“…Now consider the corresponding LI-class defined by it. It follows from the so-called torus parametrization This situation is actually met in the standard example of the Fibonacci chain: here, 4 fixed points on the torus exist, three of which correspond to generic members and the fourth to a pair of singular members, see [4] for details. To summarize, we apply Propositions 4, 3 and 1 and Theorem 1 to obtain Theorem 2 Let Λ be a regular, generic model set that is strictly inversion symmetric, and let x = x Λ be the corresponding aperiodic bi-infinite letter sequence.…”