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:
The 
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 under
quarter-cell coordinate transformations. In keeping with previous literature,
we often represent this conjugacy class with
rather than the simpler conjugate group .
Re. methods of visualizing the state-space portraits with the
4th-order isotropy subgroup quotiented out: the double-angle trick from Lorenz will not suffice here, since
we have mirror symmetry 
as well as the
rotation-about axis 
. 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
and 
, leaving us with distinct
. And it's we are most interested
in equating. – John F. Gibson 2009-03-19
