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 \mathbb{R}^n-action. There is a lower bound for M_X(T) in terms of N_X(T), namely M_X(T) \ge 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 \to \infty and is densely repetitive if M_X (T) = O(N_X(T))^{1/n} as T \to \infty. 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 \mathbb{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’.