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<- [[:chaosbook]]

====== Chapter: World in a mirror ======

(ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#discrete|World in a mirror]])  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-12//

===== Discrete symmetry desymmetrization =====

==== Quotienting the discrete translation pCf isotropy subgroup ====

From Halcrow et al. paper on pCf equilibria:

<latex>
\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}
</latex>

The <latex>R_{xz}</latex> 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 <latex>R_{xz}</latex> under
quarter-cell coordinate transformations. In keeping with previous literature,
we often represent this conjugacy class with
<latex>S = \{e, s_1, s_2, s_3\} = \{e, \sigma_z \tau_x, \sigma_x \tau_{xz},
\sigma_{xz} \tau_z\}</latex> rather than the simpler conjugate group <latex>R_{xz}</latex>.

{{gtspring2009:gibson.png?24}} Re. methods of visualizing the state-space portraits with the 
4th-order <latex>R_{xz}</latex> isotropy subgroup quotiented out: the double-angle trick from Lorenz will not suffice here, since
we have mirror symmetry <latex>(x,y,z) \to (-x,y,z)</latex> as well as the
rotation-about axis <latex>(x,y,z) \to (-x,y,-z)</latex>. 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
<latex>\tau_{xz} EQ2 to EQ2</latex> and <latex>\tau_z EQ2 \to \tau_x EQ2</latex>, leaving us with distinct
<latex>EQ2, \tau_x EQ2</latex>. And it's <latex>EQ2, \tau_x EQ2</latex> we are most interested
in equating. -- // John F. Gibson 2009-03-19//
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