Symmetries of the Web

In the theory of stochastic webs one is mainly interested
in the
symmetrical
dynamical patterns that evolve in the phase plane as a result
of the dynamics of the kicked harmonic oscillator.
The types of symmetry that can be of interest here are determined by
the symmetry groups
admitted by
the Hamiltonian (1.17).
Its first part describes a rotation with unit
angular velocity in the phase plane; as this rotation is interrupted by
the kicks stroboscopically, with time constant
,
one can
expect rotational
symmetries
through
the angle
(and integer multiples
thereof),
i.e. -fold rotation invariances.
The second part of the Hamiltonian (1.17), describing the kicks,
exhibits
translational
invariance with respect to shifts

(1.19) |

Before taking into account the characteristic properties of the kicked
harmonic oscillator (1.17),
it is useful to determine what kinds of
both
rotationally and translationally
invariant
tilings of a two-dimensional plane are possible at all
[Wey82].
See also [Lam93,LQ94,Jun95] for a more general
exposition of this topic.
The symmetry groups describing such tilings are the
*planar space groups* or
*wallpaper groups*
[GS87].^{1.4}

Consider a set of points in the plane that is invariant both with respect to translations by the arbitrary period in some direction and with respect to rotations through the angle about the points of . Those that fit into this setting are to be identified now. Depending on the value of , there may be a single or two transversal directions of translation invariance.

Let and be elements of . Submitting these points to rotations through about the points and , respectively, the points and are obtained (see figure 1.5),

and because of the stipulated symmetries of , the(1.20) |

The phase space structures with are the periodic stochastic webs introduced on page ; frequently used synonymous terms are ``uniform stochastic webs'' or ``stochastic webs with crystal symmetry''. The aperiodic stochastic webs obtained for are also referred to as ``stochastic webs with quasicrystal symmetry''.

Although the webs for
and
share their general
web-like structure, they are fundamentally different beyond the issue
of periodicity: The width of the
channels of the latter
typically
decreases with the distance from the origin of
the phase plane, whereas the periodicity of the first guarantees that
the width of the channels is the same around every mesh of the web.
Numerous examples of webs with quasicrystal symmetry can be found in the
literature; see for example [CSUZ87,SUZ88,ZSUC88,CSUZ89,ZSUC91]
and references therein.
Aperiodic webs
play an important role in the theory of
*quasicrystals* in solid state physics
[HG94,Zum97] and are related to many other fields,
for example to the problem of tilings of the plane which finds its
application both in arts (see figure 1.6 for an example)

- ...GruenbaumShephard1987.
^{1.4} *Space groups*is the general term used with respect to periodic (translation invariant) and rotation invariant patterns in arbitrary dimensions. Because of the most important application of these groups, in crystallography, they are also referred to as the*crystallographic groups*[Lax74].