In the theory of stochastic webs one is mainly interested
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
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
(and integer multiples
i.e. -fold rotation invariances.
The second part of the Hamiltonian (1.17), describing the kicks,
invariance with respect to shifts
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),
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)