On the Pisot Substitution Conjecture

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

“…, M r+ℓ,r+ℓ−1 contains at least one non-zero matrix. Such a normal form can always be achieved by a suitable permutation of the coordinates, hence by 1 A square matrix M is called decomposable if it can be brought to the block-triangular form M ′ = ( A 0 B C ) via simultaneous permutations of its rows and columns, and indecomposable otherwise [32].…”

“…However, with our present knowledge, it is still rather far to a classification in sufficient generality. While the general Pisot substitution conjecture, despite great progress in recent years (see [1] and references therein), is still open, the class of constant-length substitutions is essentially understood, at least on an algorithmic level. In fact, building on [47], Bartlett [15] presented a general method how to determine the spectral measure of maximal type computationally, for any given primitive constant-length substitution.…”

Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals

“…The Pisot Substitution Conjecture (see also the recent review Akiyama et al 2015) states the following:…”

Three variations on a theme by Fibonacci

“…The two members of the 2-cycle are locally indistinguishable and globally agree on the positive quadrant. τ, τ, τ2 and|v = τ −4 , τ −3 , τ −3 , τ −2 T ,as above normalised such that 1|v = u|v = 1. One gets lim n→∞ τ −2n M n = |v u| = P = P 2 ,…”

“…The four DPVs with the original windows of the square Fibonacci tiling, up to a global translation, with parameters as shown. The windows for the four types of control points are distinguished by colour, namely red (0), yellow (1), green(2) and blue(3). The outer boxes mark the square [−τ, τ ] 2 , with the coordinate axes indicated as well.…”