Repetitive Delone sets and quasicrystals

Abstract

This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patch-counting function N X (T ) of radius T being finite for all T . We formulate conjectures relating slow growth of the patch-counting function N X (T ) to the set X having a non-trivial translation symmetry.A Delone set X of finite type is repetitive if there is a function M X (T ) such that every closed ball of radius M X (T ) + T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with R n -action. There is a lower bound for M X (T ) in terms of N X (T ), namely M X (T ) ≥ c(N X (T )) 1/n for some positive constant c depending on the Delone set constants r, R, but there is no general upper bound for M X (T ) purely in terms of N X (T ). The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M X (T ). For example, the function M X (T ) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if M X (T ) = O(T ) as T → ∞ and is densely repetitive if M X (T ) = O(N X (T )) 1/n as T → ∞. We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure. In the reverse direction, we construct a repetitive Delone set X in R n which has M X (T ) = O(T (log T ) 2/n (log log log T ) 4/n ), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets and we propose considering them as a notion of 'perfectly ordered quasicrystals'. § Current address: . ‡ The term 'local indistinguishability class' has also been proposed for this concept. † The suffix T is to indicate that the partial order derives from inclusion of translation atlases. In [26] this partial order is considered along with a partial order ≤I derived from inclusion of isometry atlases.

A spectral sequence for theK-theory of tiling spaces

“…In addition to Kellendonk's work, several important contributions helped to build tools to prove the higher dimensional version of the Gap Labeling Theorem. Among the major contributions was the work of Lagarias [45,46,47], who introduced a geometric and combinatoric aspect of tiling through the notion of a Delone set. This concept was shown to be conceptually crucial in describing aperiodic solids [11].…”

Substitution Delone sets with pure point spectrum are inter-model sets

“…We refer the reader to [24] for more detailed definitions and to [18] for the standard concepts of discrete geometry in the aperiodic setting.…”

Aperiodic Linearly Repetitive Delone Sets Are Densely Repetitive

“…Here, the hull is the closure of its translates in the natural topology [2]. For repetitive Delone sets non-periodicity and aperiodicity are equivalent.…”

“…Conjecture (= Conjecture 1.2 a in [2]). Every aperiodic linearly repetitive Delone set is densely repetitive.…”

“…The next section recalls the necessary notation (mostly taken from [2]) and gives a proof of the conjecture.…”