Local & global aspects of symmetric dynamics

Let $\Gamma$ be a compact Lie group and suppose that $\Gamma$ acts on the finite dimensional real vector space V as a group of real linear transformations. In the sequel we refer to $(V,\Gamma)$, or just V, as a (group) representation.

Remark 2.1   Every closed subgroup $\Gamma$ of the orthogonal group O(n) defines an `orthogonal' representation $(\mbox{$\mathbb{R} $ }^n,\Gamma)$. Conversely, every representation is isomorphic to an orthogonal representation. $\diamondsuit$

Our interest lies in differential equations or vector fields which are defined on V and which are symmetric relative to the given $\Gamma$-action. Thus if

 

x' = X(x) (1)

is a smooth vector field on V, we say that X is $\Gamma$-equivariant if

$$X(\gamma x) = \gamma X(x),\;\;(x \in V, \gamma \in \Gamma).$$

The assumption of equivariance implies strong restrictions on the phase portrait of (). For example, it follows from the uniqueness of solutions of ordinary differential equations that if $\phi: (a,b) \mbox{$\rightarrow$ }V$ is an integral curve of (), then so is $\gamma \circ \phi$, all $\gamma \in \Gamma$. (Thus $\Gamma$ acts on the phase portrait by permuting maximal integral curves.)

The assumption that the group $\Gamma$ acts on V, implies that not all points in V are `equal': some are more symmetric than others. We quantify this idea of symmetry of a point by defining the isotropy group of the action at a point of V. Specifically, if $x \in V$, we define the isotropy group $\Gamma_x$ at x by

$$\Gamma_x =\{\gamma \in V \mbox{$\;\vert\;$ }\gamma x = x\}.$$

The isotropy group $\Gamma_x$ is always a closed (therefore Lie) subgroup of $\Gamma$. We think of $\Gamma_x$ as measuring the symmetry of the point x (with respect to the given action of $\Gamma$). Viewed in this way, the origin always has maximal symmetry since $\Gamma_0 = \Gamma$. We say that a point $x \in V$ has trivial symmetry if $\Gamma_x$ consists only of the identity element of $\Gamma$. It turns out that for `most' representations $(V,\Gamma)$, an open and dense set of points will have only the trivial symmetry. Thus, for a point to have nontrivial symmetry is exceptional rather than the norm.

Example 2.2  

  

Figure: Isotropy for action by D₄

In Figure , we show the symmetries of points in the plane for the standard action of the dihedral group D₄. The group D₄ consists of reflections in the lines $p,q,\ell, m$ together with the cyclic group $\mbox{$\mathbb{Z} $ }_4$ generated by rotation through $\pi/2$. Nonzero points on the lines $p,q,\ell, m$ all have isotropy group isomorphic to $\mbox{$\mathbb{Z} $ }_2$. All other nonzero points in the plane have trivial isotropy. $\heartsuit$

Using equivariance, together with the smooth dependence of integral curves on initial conditions, it is easy to see that isotropy groups are constant along maximal integral curves of (). This apparently innocuous observation about isotropy along integral curves has the significant and surprising consequence that V contains subsets which are flow-invariant for all equivariant vector fields.

Example 2.3   Suppose that X is a smooth D₄-equivariant vector field defined on $\mbox{$\mathbb{R} $ }^2$, where we take the action of D₄ given in Example . Let Abe the open wedge shaped region shown in Figure  and suppose that $x_0 \in A$. Let C be the maximal integral curve through x₀. Since x₀ has trivial isotropy, all points of C must have trivial isotropy. It follows that C is a subset of A, for otherwise Cwould have to contain points on one of the lines m, q. Hence A is a flow-invariant subset of $\mbox{$\mathbb{R} $ }^2$ for all D₄-equivariant vector fields on $\mbox{$\mathbb{R} $ }^2$. Similar arguments show that the lines $p,q,\ell, m$ and the origin are also flow-invariant subspaces. $\heartsuit$