Chapter: World in a mirror

(ChaosBook.org blog, chapter World in a mirror) — Predrag Cvitanovic 2009-02-12

Discrete symmetry desymmetrization

Quotienting the discrete translation pCf isotropy subgroup

From Halcrow et al. paper on pCf equilibria:

$\begin{equation} \label{subg4RR} R_{xz} = \{e, \sigma_x \tau_{xz}, \sigma_z \tau_{xz}, \sigma_{xz}\} = \{e,\sigma_{xz}\} \times \{e,\sigma_{z}\tau_{xz}\} \simeq S \,. \end{equation}$

The $R_{xz}$ isotropy subgroup is particularly important, as the equilibria belong to this conjugacy class, as do most of the solutions reported here. The NBC isotropy subgroup of Schmiegel and our S are conjugate to $R_{xz}$ under quarter-cell coordinate transformations. In keeping with previous literature, we often represent this conjugacy class with $S = \{e, s_1, s_2, s_3\} = \{e, \sigma_z \tau_x, \sigma_x \tau_{xz}, \sigma_{xz} \tau_z\}$ rather than the simpler conjugate group $R_{xz}$ .

Re. methods of visualizing the state-space portraits with the 4th-order $R_{xz}$ isotropy subgroup quotiented out: the double-angle trick from Lorenz will not suffice here, since we have mirror symmetry $(x,y,z) \to (-x,y,z)$ as well as the rotation-about axis $(x,y,z) \to (-x,y,-z)$ . The double-angle trick is suitable only for the latter. It would reduce the four quadrants to two, but unfortunately not in the way we would like: it would map $\tau_{xz} EQ2 to EQ2$ and $\tau_z EQ2 \to \tau_x EQ2$ , leaving us with distinct $EQ2, \tau_x EQ2$ . And it's $EQ2, \tau_x EQ2$ we are most interested in equating. – John F. Gibson 2009-03-19