Fusion: a general framework for hierarchical tilings of $$\mathbb{R }^d$$ R d

Rokhlin Dimension for Flows

“…When the topological flow R is uniquely ergodic, or, equivalently, when has uniform cluster frequencies in the sense of [LMS02] (see also [FS14,Section 3.3]), the classifying invariant consists of the topological K -groups K 0 ( ) and K 1 ( ), which are direct limits of the often easily computable topological K -groups of the Gähler approximants, together with the order of the latter group given by the Ruelle-Sullivan map K 1 ( ) → R induced by the unique invariant probability measure.…”

Topological Bragg Peaks and How They Characterise Point Sets

“…(2) The torus parametrisation π is almost everywhere injective if and only if Λ is a regular cut and project set [11]. (3) π is bijective if and only if Λ is a periodic set (has n independent periods) [11,12].…”

“…The technique of fusion [3] allows to construct modications of the Fibonacci tiling by scrambling it slightly up on each scale.…”

Meyer sets, topological eigenvalues, and Cantor fiber bundles

“…This turns out to be false. A recent example of [13], called the scrambled Fibonacci tiling, provides a counterexample. It has pure point dynamical spectrum, but none of the eigenfunctions can be chosen continuous, and hence it is not a Meyer tiling (something that can also be checked directly).…”

“…Indeed we provide first a counterexample to Lagarias' Problem 4.10 [23], namely a repetitive FLC Delone set that is pure point diffractive but not Meyer. This example can already be found in [13]: it is the scrambled Fibonacci tiling with tile lengths φ and 1. A variation of this tiling with tile lengths all equal yields an example of a repetitive Meyer set that is pure point diffractive but has non-topological eigenvalues.…”