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\vskip 3 truecm }
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\def\Om{\Omega}
\def\om{\omega}
\def\G{{\cal G}}
\def\a{\alpha}
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\def\th{\theta}
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{ \nopagenumbers
\titlea{A splitting lemma for equivariant dynamics}
% \footnote{}{May 17, 1994}
\centerline{\it Giuseppe Gaeta}
\centerline{Centre de Physique Theorique, Ecole Polytechnique}
\centerline{F-91128 Palaiseau (France)}
\centerline{and}
\centerline{Departamento de Fisica Teorica II, Universidad
Complutense}
\centerline{E-28040 Madrid (Spain)}
\footnote{}{Work partially supported by C.N.R. grant 203-01-62}
\vskip 3 truecm
{\bf Summary.} It is shown that a number of interesting and useful
results in equivariant dynamics can be seen as a consequence of a
general "splitting principle". Both the results and the principle are
actually embodied by the Michel theory of symmetry breaking.
\vfill \eject }
\pageno = 1
\vfill
\titleb{1. Introduction}
The purpose of this short note is to state (and prove) explicitely a
"splitting lemma" for equivariant dynamics; this allows to separate
the equivariant dynamics on a $G$-manifold $M$ into the "semidirect
product" of a dynamics in the $G$-orbit space $\Om = M/G$ and a
dynamics on $G$.
Some readers will notice that the result given below, as well as the
construction employed and the (idea of the) proof, just follow from
the statement and the proof of Michel theorem [1] on the geometry of
symmetry breaking (see also [2,3] for discussion and applications of
this, and [4,5] for extensions). Thus, it could seem not so useful to
give an explicit statement and proof of the present lemma.
On the other side, several results have appeared after [1] was
published, which could be considered as corollaries - or special cases
- of Michel theorem, and which were obtained with no reference to
[1]. Among these, we mention the "Equivariant Branching Lemma" [6]
and the "Reduction Lemma" [7], which play a key role in the
developement (and applications) of Equivariant Bifurcation Theory
[8,9], as discussed in [10]. Also, some more recent results can be
seen as natural in the light of the splitting lemma; among these we
mention the beautiful work by Krupa [11] on bifurcation from a
relative equilibrium, and the very recent contribution by Chossat and
Koenig [12].
Thus, with the present note we aim on the one side at pointing out
how these different results are manifestations of the same basic
facts; and on the other at pointing out how the Michel theory
encompasses and unifies these different aspects.
Michel theory, developed in the context of elementary particle theory,
has often been overlooked in the context of equivariant dynamics (as a
significant exception to this, we mention the far-reaching works by
Field and Richardson [13]); this fact is presumably due to lack of
communication among different fields, and among Physics and
Mathematics communities in general; it is hoped that the present note
will call the attention of the readers to this unifying and beautiful
theory.
We give here an elementary and self-contained derivation of the
splitting lemma (SL in the following); a more complete discussion,
considering also the relation of the SL to the results mentioned above
and to Michel theory, will be given elsewhere [14]; we also postpone
the discussion of more general cases (e.g. non-compact Lie groups) to
there.
\eject
\titleb{2. Group action}
Let $M$ be a smooth (i.e. ${\cal C}^\infty$) real manifold, which we
will think as embedded in $R^n$ for ease of notation. Let $G$ be a
connected compact Lie group acting smoothly in $M$. We denote by $\Om
= M/G$ the $G$-orbit space (or orbit space {\it tout court}) for the
action of $G$ on $M$, i.e. the space of equivalence classes of points
of $M$ for the (equivalence) relation of "lying on the same $G$-orbit";
elements od $\Om$ will be denoted as $\om$, while we write $\om [x]
\sse M$ for the set of points of $M$ on the same $G$-orbit as $x$:
$$ \om [x] := \{ y \in M ~:~ y = gx ,~ g \in G \} \equiv \{ Gx \}
\eqno(1) $$
At any point $x \in M$, let us consider the invariant splitting of the
tangent space $T_x M$ given by
$$ T_x M = T_x \om [x] \oplus N_x \om [x] \eqno(2) $$
where $T \om [x]$ (respectively $N \om [x]$) is the tangent bundle
(respectively the normal bundle) to $\om [x]$.
Let us consider the Lie algebra $\G$ of $G$; this will have
generators $ \{ \eta_1 , \dots , \eta_r \}$ which we will write in a
concrete realization for the $G$-action on $M$ as
$$ \eta_j = \sum_{i=1}^n \phi_{(j)}^i (x) \cdot {\pa \over \pa x^i }
\eqno(3) $$
so that $T_x \om [x]$ is the span of $\{ \phi_{(1)} (x) ,
\dots , \phi_{(r)} (x) \}$ (notice that these are not necessarily
linearly independent at all points $x \in M$).
To any point $x \in M$ we can associate an {\it isotropy subgroup}
$G_x$ and a {\it conjugated fixed space} $M^x$ by
$$ G_x := \{ g \in G ~:~ gx = x \} \eqno(4) $$
$$ M^x := \{ y \in M ~:~ gy = y ~ \forall g \in G_x \} \equiv \{ y \in
M ~:~ G_x \sse G_y \} \eqno(5) $$
Notice that for points on the same $G$-orbit we have
$$ G_{gx} = g G_x g^{-1} \eqno(6) $$
$$ M^{gx} = g ( M^x ) \eqno(7) $$
Subgroups of $G$ which are conjugated in $G$ should be seen as
equivalent; based on this observation one is naturally led to Michel
theory, i.e. to consider the {\it isotropy stratification} of $M$ and
$\Om$ [1]; For this we refer to [1-3]; we will also discuss this
in [14], while here we choose to limit ourselves to simpler
considerations.
Let us now focus on the orbit space $\Om$; by a theorem of Hilbert
[15] (for polynomials) and Schwartz [16] (for ${\cal C}^\infty$
functions), there is a {\it minimal integrity basis} (MIB) - in
general, non unique - of $G$-invariant ${\cal C}^\infty$ functions
(respectively, polynomials) $\{ I_1 (x) , ... , I_h (x) \}$, such that
any $G$-invariant ${\cal C}^\infty$ function (respectively,
polynomial) $F(gx)=F(x)$ can be written as a ${\cal C}^\infty$
function (respectively, a polynomial function) of the $I_j$'s, i.e.
$F(x) = {\widetilde F} (I_1 (x) , ... , I_h (x) )$ [1-3,15,16].
The orbit space $\Om$ is then a semialgebraic manifold, diffeomorphic
to a semialgebraic set $S_\Om \sse R^h$.
An invariant function $F : M \to R$, $F (gx)=F(x)$, can also be seen as
a function on the $G$-orbit space; we denote the corresponding function
as $F_\om : \Om \to R$; we also write $\om : M \to \Om$ the function
which associates to $x \in M$ the corresponding orbit as a point in
$\Om$, so that $F \equiv F_\om \cdot \om$. If we denote by $I : M
\to R^h$ the map associating to $x \in M$ the values $\{ I_1 (x) , ...
, I_h (x) \}$ of functions in the chosen MIB, and by $\om : M \to \Om $
the map associating to a point $x \in M$ the corresponding element of
$\Om$, we have the following diagram:
$$ \matrix{
M & \mapright{I} & R^n \cr
\mapdown{\om} & & \mapdown{{\widetilde F}} \cr
\Om & \mapright{F_\om} & R \cr} $$
In practice, invariant functions $I(x)$ are determined by asking
$\eta \cdot I = 0$ $\forall \eta \in \G$; using (3), this leads to a
system of linear PDEs of the form
$$ \sum_{i=1}^n \phi_{(j)}^i (x) {\pa I \over \pa x^i } = 0 ~~~~~ j =
1,...,h \eqno(8) $$
or equivalently to the characteristic equations [17,18]
$$ {d x^1 \over \phi_{(j)}^1 } = ... = {d x^n \over
\phi_{(j)}^n } ~~~~~ j = 1, ..., h \eqno(9) $$
\titleb{3. Equivariant dynamics}
Let us now consider a (smooth) dynamical system defined on $M$; we
write this in coordinate form as
$$ {\dot x} = f(x) ~~~~ f : M \to TM \eqno(10) $$
or, more intrinsically, we consider a vector field $\psi$ on $M$, whose
coordinate expression is $ \psi = \sum_i f^i (x) \pa / \pa x^i$. We
consider $G$-equivariant vector fields: this means
$$ [ \psi , \eta ] = 0 ~~~~ \forall \eta \in \G \eqno(11) $$
or, in the coordinate form (10),
$$ f (gx ) = (D_g f ) (x) ~~~~ \forall g \in G \eqno(12) $$
where $D_g$ denotes the action of $g$ on $TM$.
Let us consider the time evolution of a $G$-invariant quantity $I(x)$
under the flow of $\psi$:
$$ {\dot I} \equiv { d I \over d t} = ( \psi
\cdot \grad ) I = f^i (x) {\pa I \over \pa x^i } \eqno(13) $$
In particular we can consider $I$ to be one of the $I_j$ in the
chosen MIB; then the time evolution of any $G$-invariant function $F$
is given - in the notation introduced above - by
$$ {\dot F} (x) = \sum_{j=1}^h {\pa {\widetilde F} \over \pa I_j} {\dot
I}_j \eqno(14) $$
It is just a simple remark that the ${\dot I_j}$ are themselves
$G$-invariant quantities: thus, there exist functions $\a_j : R^h \to
R$ such that
$$ {\dot I}_j = f^i (x) {\pa I_j \over \pa x^i } := \a_j
(I_1 , ... , I_h ) \eqno(15) $$
In other words, $\psi$ induces an {\it
autonomous} dynamical system - i.e. a vector field $A = \a_j \pa / \pa
I_j$ - on $S$, and thus on $\Om$. This is also {\it smooth} since all
the mappings involved are smooth [1-3,15,16], and since singular
points of $S$ turn out to correspond necessarily to invariant sets and
zeroes of the vector field [1-3,14].
\bigskip
{\bf Lemma I.} {\it A smooth $G$-equivariant dynamical system on the
$G$-manifold $M$, induces a smooth dynamical system on the orbit space
$\Om = M/G$.}
\bigskip
Given a manifold $M$, we can think to choose a subset $M_0 \sse M$
such that each $G$-orbit in $M$ intersects $M_0$ precisely once (so
that $\Om$ is isomorphic to $M_0$); then any point $x \in M$ can be
described in terms of two coordinates $y \in M_0$ and $g \in G$ just
as
$$ x \simeq (y,g) \Leftrightarrow x = gy ~~~~ y \in M_0 \eqno(16) $$
(notice $g$ in (16) is only defined up to $G_y$).
Let us decompose the flow of $x(t)$ into the direction tangent to the
$G$-orbit $\om [x(t)]$ and into that orthogonal to the same $G$-orbit,
see (2). Corresponding to this, the flow of $\psi$ will be decomposed
into a flow $A$ in the orbit space $\Om$, and a flow along $G$-orbits
or, equivalently, a flow on $G$. Notice that the latter is only
defined up to elements of $G_{x(t)}$ (which is a kind of gauge
invariance).
The fact we define in this way a {\it smooth flow} on $G$ can be seen
as a consequence of general theorems in Bredon [17] (chapt.VI), and
depends crucially on our smoothness assumptions. Again, it is
important in this respect that orbits with greater isotropy subgroups
are invariant sets for the flows [1-3,13], which leads us again to
rely on Michel theory. The smoothness of the flow on $G$ also amounts
to the possibility of choosing $M_0 \sse M$ above, as a smooth
(embedded) submanifold of $M$.
Notice that the flow on $G$ will, in general, depend on the orbit we
are on. In other words, let us choose coordinates $\{ \th_1 , ... ,
\th_k \}$ on $G$; then the flow on $G$, i.e. the time evolution of the
$\th_j$'s, will depend on the $I_j$'s, namely will be described by
equations of the form
$$ {\dot \th}_j = \b_j ( \th_1 , ... , \th_k ; I_1 , ..., I_h )
\eqno(17) $$
We will summarize our discussion as follows:
{\bf Splitting Lemma.} (Invariant formulation) {\it Let $M$ be a
smooth $G$-manifold, with $G$ a smooth compact connected Lie group
acting smoothly on $M$; let $\psi$ be a smooth $G$-equivariant vector
field on $M$. Let $\Om = M/G$ be the $G$-orbit space on $M$. Then,
$\psi$ induces an autonomous vector field $A$ on $\Om$, and a
$\Om$-dependent flow on $G$.}
{\bf Splitting Lemma.} (Coordinate formulation) {\it Let $M \sse R^n$
be a smooth manifold, $G$ a smooth compact connected Lie group acting
smoothly on $M$. Choose coordinates $\{ \th_1 ,..., \th_k \}$ in $G$,
and coordinates $\{ I_1 ,..., I_h \} $ in $\Om = M/G$ (i.e. a MIB for
$G$). Consider a smooth $G$-equivariant dynamical system in $M$,
$$ {\dot x} = f(x) ~~~~ f:M \to TM $$
$$ f (gx) = (D_g f) (x) ~~~~ \forall g \in G $$
Then, the evolution of the $\{ \th ; I \}$ is governed by a "triangular system"
$$ \eqalign{ {\dot I} =& \a (I) \cr {\dot \th} =& \b (I, \th ) \cr } $$ }
\titleb{4. Remarks and extensions}
{\bf Remark 1.} Symmetry considerations do also grant the invariance
of specific submanifolds of $M$ under $G$-equivariant flows: e.g., the
manifolds $M^x$ introduced above are invariant, so that one can
restrict to such invariant manifolds, simplifying considerably the
study of the dynamics. In particular, in this case only the subgroup
$N_G (G_x )\sse G$ is accessible to the dynamics (here $N_G (H)$ is the
normalizer of $H \sse G$ in $G$), and only $D_G (G_x ) := N_G (G_x ) /
G_x$ acts effectively in $M^x$. Therefore, for a given initial datum
$X(0) = x_0$, we restrict to dynamics in $\Om$ and flow on $D_G
(G_{x_0} )$.
{\bf Remark 2.} In some cases - e.g. for $H$ a maximal isotropy
subgroup of $G$ - the group $D_G (H)$ is particularly simple [17],
and we can provide stronger results [10].
{\bf Remark 3.} Invariant subsets are also granted to exist in $\Om$,
due to symmetry. This can be seen by using Michel's theory of isotropy
stratifications [1-3], which leads to a hierarchy of invariant
subsets in $\Om$. It is in this way that one obtains e.g. the
Equivariant Branching Lemma [6]
{\bf Remark 4.} If the dynamical system in $\Om$ admits fixed points
(respectively, periodic or multiperiodic orbits), in correspondence to
these one has to study autonomous flows (respectively, flows with
periodic or multiperiodic coefficients) in $G$, or more precisely in
$D_G (G_{x_0})$. Again, Michel's theory can in some cases grant the
existence of fixed points for the flow in $\Om$ on the basis of
symmetry considerations alone.
{\bf Remark 5.} In the situation mentioned in remark 4, one can use a
number of powerful tools (momentum mapping, coadjoint orbits theory),
as we are reduced to a (simple) dynamics on a Lie group. Notice that
for the non-stationary (in $\Om$) case, we could have troubles if in
the presence of a nontrivial topology for $\Om$. It is conjectured
([17], chapt.I) that this is never the case for $M = R^n$ or $R=D^n$.
{\bf Remark 6.} Notice that fixed points for the dynamics in $\Om$
correspond to {\it relative equilibria} for the overall dynamics in
$M$.
\titleb{5. Simple examples}
{\bf Example 1.} Let $M = R^2$, $G = SO(2)$ acting in the standard
way. We pass to polar coordinates $( \rho , \th )$ (with $\rho = (x^2
+ y^2 )/2$), which are also coordinates for $\Om$ and for $G$
respectively. Indeed, any $G$-equivariant system is written as
$$ \eqalign{
{\dot x} = & a (\rho ) x - b (\rho ) y \cr
{\dot y} = & b (\rho ) x + a (\rho ) y \cr } $$
which yields
$$ \eqalign{
{\dot \rho} =& \rho a( \rho ) \cr
{\dot \th } =& b (\rho ) \cr } $$
{\bf Example 2.} Let $M = R^4 $, $G = SU(2)$ acting by the
realification of its defining representation. Then there are three
matrices $H_1 , H_2 , H_3 $ (generating a representation of the
$SU(2)$ algebra themselves) [19] such that any $G$-equivariant system
is written as
$$ {\dot x} = a (\rho ) x + \sum_{i=1}^3 b_i ( \rho ) H_i x $$
and we have
$$ {\dot \rho} = \rho a (\rho ) $$
while the flow along $G$ is given simply by
$$ {\dot x} = \sum_{i=1}^3 b_i (\rho ) H_i x $$
{\bf Example 3.} Let $M$ be the solid torus, on which we take
coordinates $\rho \in [0,1]$, $\th \in S^1$, $\phi \in S^1$ (say, for
pictorial representation, with $\th$ going around the equator of the
torus); let $G = SO(2) \times SO(2)$ acting through $\th$ and $\phi$
independent rotations, so that $\rho$ parametrizes $G$-orbits ($\rho
=0$ corresponds then to an exceptional orbit being a circle $S^1$, all
the other $G$-orbits being tori $T^2 \simeq G$). In these
coordinates, $G$-equivariant systems are written as
$$ {\dot \rho} = a (\rho ) ~~;~~ {\dot \th} = b(\rho ) ~~;~~ {\dot
\phi} = c( \rho ) $$
Let us introduce explicitely a control parameter $\l$, so that ${\dot
\rho} = a (\rho , \l )$, etc. We make standard bifurcation
assumptions, i.e. we assume that $ \arl < 0$ for $\rho$ large enough
and for any $\l$, and that at $\rho = 0$ we have $ \pa \arl / \pa \rho
< 0$ for $\l < 0$, and $\pa \arl / \pa \rho > 0 $ for $\l > 0$ (Michel
theorem ensures $ a (0,\l ) =0 $ $\forall \l$). Also, we assume that
$b,c$ have definite signs for $\l \simeq 0$, $\rho \simeq 0$ (this a
genericity requirement), e.g. $b(0,0) > 0 $, $c (0,0) > 0$.
With these, we have a (symmetry breaking) bifurcation from the
relative equilibrium $\rho =0$ for $\l =0$. In this bifurcation, the
symmetry $G_0 \subset G$ corresponding to $\phi$ rotations is broken;
the splitting lemma guarantees that the bifurcating flow is given by a
small deformation of the flow on $S^1$ times a flow on $G_0$; this is
indeed the kind of results obtained by Krupa [11].
\vfill \eject
\titleb{References}
\parskip=5pt
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\bye