Proximality and pure point spectrum for tiling dynamical systems

Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

“…We have seen that numerical results for the ground states in Amman-Beenker and Penrose tilings yield purely real values for the factor λ i in (18). This is rather unexpected since this factor plays the role similar to that of BlochFloquet multiplier.…”

“…Then the function ∆(x) = D(x 1 + x, x 2 + x) characterizes what is called in the physical literature the response of the structure to an infinitesimal phason shift. The points x 1 and x 2 are proximal for the dynamical system (Ω, R d ) [18], that is inf (∆(x)) = 0. However, the stability of quasicrystal requires unboundedness and connectedness of the set {x, ∆(x) > δ} ⊂ R d for some δ > 0 since otherwise no local interaction could enforce a globally coherent choice between the patterns corresponding to x 1 and x 2 .…”

On the Pisot Substitution Conjecture

“…m-to-1 for some m ∈ N ∪ {∞}, whereT is a torus or solenoid and the R n -action is a Kronecker action. The number m = cr(Φ) is the coincidence rank of Φ ( [28]). Also, by [113,29,21], for one-dimensional primitive tile substitutions Φ, cr(Φ) < ∞ if and only if the expansion λ for Φ is a Pisot number.…”

“…It is a fundamental theorem of Veech [119] that the structure relation for the maximal equicontinuous factor map g is regional proximality: tilings T , T ∈ Ω Φ are regionally proximal, T ∼ rp T , if and only if, for each > 0 there are S, S ∈ Ω Φ and v ∈ R n so that: (i) d(T , S) < , (ii) d(T , S ) < , and (iii) d(S − v, S − v) < . (For a general discussion of regional proximality in tiling spaces see [28].) In the context of Theorem 7.7 one can show that T ∼ gs T =⇒ T ∼ rp T .…”

Topological Bragg Peaks and How They Characterise Point Sets

“…Furthermore cr = mr whenever Ω Λ contains an element which is not proximal to any other element. Primitive Meyer substitution tilings yield examples for which cr = mr ≤ M r < ∞ [13]. The ThueMorse substitution has cr = 2.…”