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 $2\pi/q$, one can expect rotational symmetries through the angle $2\pi/q$ (and integer multiples thereof), i.e. $q$-fold rotation invariances. The second part of the Hamiltonian (1.17), describing the kicks, exhibits translational invariance with respect to shifts

$$x \; \longmapsto \; x+2n\pi, \quad n\in\mathbb{Z},$$ (1.19)

in $x$-direction. As a result, one has to look for phase space patterns that show both these rotational and translational symmetries at the same time, where (by the rotations) the translation invariance is not necessarily restricted to the $x$-direction any more.

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

Let $A$ and $B$ be elements of ${\mathcal P}$. Submitting these points to rotations through $\alpha$ about the points $B$ and $A$, respectively, the points $A'$ and $B'$ are obtained (see figure 1.5),

and because of the stipulated symmetries of ${\mathcal P}$, the crystallographic condition [Lax74,Che89]

$$\cos\alpha \; = \; \frac{1-n}{2} \quad \mbox{with % a suitable $n\in\mathbb{Z}$}$$ (1.20)

must be satisfied. This implies that $n\in\{-1,0,1,2,3\}$ and therefore $\alpha=2\pi/q$ with $q\in {\mathcal Q}$. The case $q=1$ is trivial in the sense that here no rotation is performed at all. Similarly, $q=2$ is not really interesting as the corresponding rotations are through the angle $\pi$, such that the resulting pattern in the plane just consists of parallel lines of equidistant points. (Compare with the discussion of the maps $M_1$ and $M_2$ in subsection 1.1.3. -- In planar crystallography the corresponding symmetry groups are called strip groups or frieze groups.) In order to obtain webs with translational symmetry in two transversal directions and nontrivial rotational symmetries by means of the kicked harmonic oscillator I therefore restrict the larger part of the following investigation to kicks with the period

$$T=\frac{2\pi}{q}, \quad q \in \tilde{{\mathcal Q}} := \{ 3,4,6 \} \subset {\mathcal Q}.$$ (1.21)

The phase space structures with $q\in {\mathcal Q}$ 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 $q\not\in {\mathcal Q}$ are also referred to as ``stochastic webs with quasicrystal symmetry''.

Although the webs for $q\in {\mathcal Q}$ and $q\not\in {\mathcal Q}$ 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)

and science [Kep19,GS87]. The idea to explain symmetries of a planar pattern by identifying an appropriate model system which dynamically generates that pattern is originally due to KEPLER [Kep11].

Footnotes

...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].