Next: Quasi-periodic sphere-packings Up: Maximizing the packing density Previous: Introduction

# Sphere packings and dynamical systems

A r-sphere packing S is a countable set of points in such that the minimal distance between two points is . Let be the set of all sphere packings. Define a metric on by , where is the smallest number such that on the ball for some . With this metric, which is adapted from the metric for tilings, is a complete metric space. Let act on by translations . Given a specific sphere packing S, we can look at the closure of the orbit . If compact in , we get a dynamical system , where is a compact metric space on which acts by homeomorphisms. Such a system is called minimal or almost periodic, if for every , the orbit is dense in , it is called uniquely ergodic, if there exists exactly one - invariant measure on , strictly ergodic, if it is uniquely ergodic and minimal.

Remarks.
1) A metric on , with respect to which the subset of r-packings in becomes compact is described in [5]. For our purposes, the stronger metric d is good enough.
2) In the statistical mechanics literature, a sphere packing is also called a configuration with hard core restriction.
3) By normalization of the distance in , we could assume that a sphere packing has radius r=1. We keep the additional parameter r since we are interested in packings .

We call a sphere packing S rational, if all points of S belong to a d-dimensional lattice , where U is an invertible -matrix. On a rational sphere packing, there is a natural -action, by identifying with the maximal subgroup of which leaves the lattice invariant. The packing S consists then of a subset of the lattice and every rational sphere packing defines so a subshift . This set is is invariant under the -action. A rational sphere packings is called strictly ergodic, if the dynamical system is strictly ergodic. If every shift of the -action is periodic, S is called periodic,

Remarks.
1) Clearly, a periodic packing is rational and also strictly ergodic.
2) Periodic packings are dense in since we can periodically continue a given packing outside a given box. Periodic packings are also dense in the set of rational packings.
3) The name almost periodic which stands as a synonym for minimal has no relation with the usual almost periodicity of functions or sequences. The expression almost periodic is however widely used in the topological dynamics and mathematical physics literature.

The lower and upper densities of a sphere packing S are defined as

where . If , then is called the density of S.

Remarks.
1) The above lemma is well known and there are other proofs using more theory. The result follows for example also from a multi-dimensional version of Birkhoff's ergodic theorem (see [4] Chapter VIII). The proof given here uses only lightest tools.
2) There exists a dense set of rational packings in which have no density. Proof. Consider a periodic packing S of radius r having density . Take an other periodic packing S' of radius r which has density . Take a first cubic box centered at zero and fill it with spheres of radius r according to the first packing. Take a second larger cubic box and fill with spheres according to the packing in S'. Make so large that the density of the packing in this finite box is smaller than . Take a box and fill with spheres according to make so large that the density in the box is larger than . Continue inductively so that the finite volume densities are alternatively below and above .
Given , we can make so large that the distance between the original packing and the modified packing is smaller than .
3) We are forced to define the packing problem on a subclass of packings since the density is not a continuous function on the set of all packings for which the density exists: take such a packing S and define a sequence of packings obtained from S by deleting all balls in distance less than n from the origin. The packings have all the same density but converges to the , which is a packing with zero density.

Next: Quasi-periodic sphere-packings Up: Maximizing the packing density Previous: Introduction

Oliver Knill
Mon Jun 22 17:57:55 CDT 1998