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--- chaosbook:discrete [2009/02/12 15:17]
predrag
+++ chaosbook:discrete [2010/02/02 07:55] (current)
@@ Line 7: / Line 7: @@
 ===== 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
+th-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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