%\magnification=1200
\parskip=10pt
\parindent=0pt
\def\R{{\bf R}}
\def\all{\forall}
\def\V{{\cal V}}
\def\M{{\cal M}}
\def\grad{\nabla}
\def\a{\alpha}
\def\b{\beta}
\def\eps{\varepsilon}
\def\phi{\varphi}
\def\s{\sigma}
\def\S{\Sigma}
\def\th{\theta}
\def\Th{\Theta}
\def\om{\omega}
\def\Om{\Omega}
\def\pa{\partial}
\def\der{{\rm d}}
\def\sse{\subseteq}
\def\ss{\subset}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\~#1{{\widetilde #1}}
\def\^#1{{\widehat #1}}
\def\=#1{{\bar #1}}
\def\EOS{}
\def\section#1{ \bigskip \bigskip {\bf #1} \bigskip }
\newcount\notenumber
\def\clearnotenumber{\notenumber=0\relax}
\def\fnote#1{\advance\notenumber by 1
\footnote{$^{\the\notenumber}$}{\petit #1}}
{\nopagenumbers
~ \vskip 5 truecm
\centerline{\bf Splitting equivariant dynamics}
\vskip 3 truecm
\centerline{\it Giuseppe Gaeta}
\centerline{Centre de Physique Th\'eorique, Ecole Polytechnique}
\centerline{F - 91128 Palaiseau (France)}
\centerline{and}
\centerline{Departamento de Fisica Teorica II, Universidad Complutense}
\centerline{E - 28040 Madrid (Spain)}
\vskip 2 truecm
{\bf Summary.} We prove that any dynamical system on a $G$-manifold $M$
which is equivariant under the $G$ action, can be decomposed into the
semidirect product of an autonomous dynamics in the $G$-orbit space
$\Om = M / G$, and a dynamics (depending on the $G$-orbit) on $G$. This
result is actually a corollary of Michel theorem [1] on the geometry of
symmetry breaking, and uses the same ingredients for the proof. It permits
to unify a number of known and useful results in the literature, as discussed
here.
\vfill
Work supported in part by italian C.N.R. under grant 203.01.62
\eject}
\pageno=1
\section{1. Introduction}
Let $M$ be a smooth (${\cal C}^\infty$) manifold, on which the smooth
action of a compact Lie group $G$ is defined; i.e., $M$ is a smooth
$G$-manifold. For ease of notation, we will consider $M$ to be a smooth
submanifold of $\R^N$, but our construction and results will not depend on
this assumption.
Let us consider the algbera $\V$ of $G$-invariant real functions on $M$, i.e.
$$ \V = \{ V(x) ~:~ V:M \to \R ~;~ V(gx)=V(x) ~ \all g \in G \} \eqno(1) $$
and the algebra $\M$ of smooth $G$-equivariant vector fields on $M$, i.e.
$$ \M = \{ \phi : M \to TM ~:~ \phi = f(x) \grad ~;~ f(gx) = (D_g f) (x) ~
\all g \in G \} \eqno(2) $$
Here we have denoted the transformed of $x$ by $g \in G$ (through the
action defined on $M$) simply by $gx$, as we consider only one $G$ action;
and the action of $g$ on $TM$ by $D_g$.
The set $\M$ is a Lie algebra once equipped with usual commutator of vector
fields $[\phi , \psi ] = \phi \psi - \psi \phi$; we could equivalently
consider the algebra
$${\cal E} = \{ f:M \to \R^N ~;~ f(gx) = (D_g f) (x) ~ \all g \in G \}
\eqno(2') $$
in which case the Lie algebra operation is the Lie-Poisson bracket $\{ f,g
\} = (f \cdot \grad ) g - (g \cdot \grad ) f$.
We want to consider $G$-equivariant dynamical systems on $M$,
$$ {\dot x} = f(x) ~~~~ f \in {\cal E} \eqno(3) $$
A subclass of these, of special interest, is given by gradient flows, i.e.
by the case
$$ {\dot x} = f(x) = \grad V (x) ~~~~ V \in \V \eqno(4) $$
In this case, we would be mainly interested in fixed points of the flow,
i.e. in critical points of $V$; indeed, these represent asymptotic
solutions to the dynamical system.
In order to study (3), (4) we will now recall a construction due to L.
Michel [1] (see
also [2-5]), based on the geometry of $G$-action in $M$, and developed in
the frame of
variational problems; we will then show that this gives as a corollary a
nice and
useful splitting lemma for the general case of equivariant dynamics (3), which
represents the main result of this note. The remaining parts of this paper
will be
devoted to discuss applications of the splitting lemma, and to show how it
permits to
unify a number of known (and, again, useful) results in equivariant
dynamics and in
particular in equivariant bifurcation theory.
%\vfill \eject
\section{2. The orbit space}
Let $\om [x] $ be the $G$-orbit through $x$ in $M$, i.e.
$$ \om [x] = \{ y \in M ~:~ y = gx ~,~ g \in G \} = \{ Gx \} \eqno(5) $$
Clearly, belonging to the same $G$-orbit is an equivalence relation in $M$,
and we can consider the space of such equivalence classes or, the {\it
orbit space}
$$ \Om = M/G \eqno(6) $$
Results on the geometry of orbit space are collected e.g. in [6,7] (see
also [2,4,8]);
here we will just recall the results which are useful for the present
discussion.
{\bf Lemma I.} Let $G$ be a compact Lie group and $M$ a smooth
$G$-manifold; then $\Om
= M/G$ is a semialgebraic manifold. \EOS
We recall that an {\it integrity basis} for the $G$-action on $M$ is a set
$\{ \th_1 ,
... , \th_k \}$ of functions $\th_i \in \V$ such that any smooth
(algebraic) function
$V(x) \in \V$ can be written as a smooth (algebraic) function $\~V $ of the
$\th$'s,
$V(x) = \~V (\th_1 (x) , ... , \th_k (x) )$. The integrity basis is called {\it
minimal} if no one of the $\th$'s can be dropped still having an integrity
bases.
{\bf Lemma II.} If $G$ is a compact Lie group acting smoothly in $M$, the
algebra of
smooth (algebraic) invariant functions on $M$ admits a finite dimensional
minimal
integrity basis. \EOS
{\bf Lemma III.} Let $\{ \th_1 , ... , \th_k \}$ be a minimal integrity
basis for the
$G$-action in $M$. Then, each $G$-orbit belongs to one common level set of
$\{ \th_1 ,
... , \th_k \}$, and conversely for any common level set of $\{ \th_1 , ...
, \th_k \}$
in $M$ there is at least a $G$-orbit whose closure coincides with the
common level set.
\EOS
These results were proved by Hilbert [9] for the algebraic case, and extended by
Schwarz [10] to the smooth case (see also [11]). The statement of lemma III is
sometimes summarized by saying that "invariants do separate orbits".
We will denote by $\th : M \to \R^k$ the mapping which associates to a
point $x \in M$
the values $\th_i (x)$ of the functions of a minimal integrity basis. Let
us denote by
$\~M$ the image of $M$ under $\th$, i.e. $ \th (M) = \~M \sse \R^k$.
Clearly, $\Om$ is
then diffeomorphic to $\~M$, and can therefore be thought as a semialgebraic
submanifold of $R^k$.
{\bf Lemma IV.} If $G$ is a compact Lie group acting smoothly in $M$, the
orbit space
$\Om = M/G$ is diffeomorphic to a semialgebraic manifold in $\R^k$, where
$k$ is the
dimension of a minimal integrity basis $\{ \th_1 , ... , \th_k \}$ for the
$G$ action;
this semialgebraic manifold is $\~M = \th (M)$. \EOS
Let us now go back to the dynamical system (3); this induce a dynamical
system in $\Om$. Indeed, let us consider the evolution of the invariant
functions $\th_i (x)$ when $x(t)$ evolves according to (3), i.e. the
evolution of $\th$'s along solutions to (3). In this case,
$$ { \der \th \over \der t} = ( f (x) \cdot \grad ) \th (x) \eqno(7) $$
Now, $\th$ is $G$-invariant, and the operator $( f \cdot \grad )$ is also
$G$-invariant, as it is the contraction of a contravariant and a covariant
object, so that the r.h.s. of (7) is an invariant function, and therefore,
according to Lemma III, can be written in terms of the $\th$'s.
Equivalently, the time derivative of the $\th$'s is clearly also
$G$-invariant, and therefore can be written in terms of the $\th$'s
themselves. Therefore, we can rewrite (7) as
$$ {\dot \th} = \~f (\th ) \eqno(8) $$
We have thus proved that the dynamical system (3) on $M$ projects to a dynamical
system (8) in $\Om \simeq \~M$
We can therefore split the dynamics in two parts: an autonomous part on
$\~M \simeq
\Om$, and an orbit-dependent dynamics on group variables (which does {\it
not} mean
"tangent to group orbits"). It should be mentioned that a great deal of this
construction survives when we consider non-compact and/or infinite
dimensional Lie
groups $G$ and smooth (Banach) manifolds $M$ (we need Banach manifolds to
define an
equivariant retraction [1,2,4,8]); we will not discuss this point, and the
limitations
of such an extension, here; the interested reader is referred e.g. to [4,6,7].
We will now discuss how the present construction is related to the geometry of
$G$-action in $M$; this discussion will also make clear the reason why we
employed the
term "group variables".
\section{3. Stratification}
To any point $x \in M$ we associate its isotropy group $G_x \sse G$ by
$$ G_x = \{ g \in G ~:~ gx=x \} \eqno(9) $$
Let us consider the equivalence relation between subgroups of G given by
conjugacy in $G$; i.e., if $H \sse G$,
$$ C[H] = \{ K \sse G ~:~ K = g H g^{-1} ~,~ g \in G \} \eqno(10) $$
This equivalence relation induces an equivalence relation in $M$, known as
{\it isotropy stratification}; the equivalence classes - or {\it strata}
(latin for layers) - being defined as
$$ \s [x] = \{ y \in M ~:~ G_y \in C [G_x ] \} \eqno(11) $$
Since $y=gx$ implies $G_y = g G_x g^{-1}$, we have that points on the same
$G$-orbit
belong to the same stratum, or $$ \om [x] \sse \s [x] \eqno(12) $$
The isotropy stratification of $M$ does therefore induce an isotropy
stratification on
$\Om$; indeed, let us denote
$$ C [\om ] = C [G_x ] ~,! x \in \om \eqno(13) $$
the {\it isotropy type} of the orbit $\om$; the strata are then defined by
$$ \S [\om ] = \{ \nu \in \Om ~:~ C [\nu ] = C [\om ] \} \eqno(14) $$
Let us now consider, for any subgroup $H \sse G$, the {\it fixed space}
$$ M^H = \{ x \in M ~:~ gx=x ~ \all g \in H \} \eqno(15) $$
In particolar, if $H = G_x$, we will just write $M^x$ for $M^{G_x}$. In
this case, (15) reads also
$$ M^x = \{ y \in M ~:~ G_y \sse G_x \} \eqno(16) $$
Notice also that we have $M^{gx} = g \( M^x \) $.
It is clear that for any point $x \in M$, $\om [x]$ is transversal to $M^x$
(unless $\om [x] = x$, i.e. $G_x = G$). More precisely, the tangent space
$T_x M$ admits the following decomposition:
$$ T_x M = T_x \om [x] \oplus N_x \om [x] \eqno(17) $$
$$ N_x \om [x] = T_x M^x \oplus N_x^{(1)} \om [x] \equiv
N_x^{(0)} \om [x] \oplus N_x^{(1)} \om [x] \eqno(18) $$
where $N_x \om [x] \sse T_x M$ is the normal subspace to $\om [x]$
in $x$, and $N_x^{(0)} , N_x^{(1)} $ denote the invariant and
non-invariant (under $G_x$) parts of $N_x \om [x]$. Notice that
(17),(18) are $G_x$-invariant decompositions, by definition, and are
also equivariants: i.e.,
$$ \eqalign{ T_{gx} \om [gx ] =& D_g \( T_x \om [x ] \) \cr T_{gx }
M^{gx} =& D_g \( T_x M^x \) \cr T_{gx} N_{gx}^{(1)} \om [gx] =& D_g
\( N_x^{(1)} \om [x] \) \cr } \eqno(19) $$
Indeed, the first two of
these follow immediately from the properties of $\om [x]$ and $M^x$;
the third one follows from these and (17),(18).
One can easily prove [1,2,4,6,7] that
{{\bf Lemma V.} \hfill $T_x \s [x] = T_x \om [x] \oplus T_x M^x
\equiv T_x \om [x] \oplus N_x^{(0)} \om [x]$ \hfill (20)}
Moreover, it can be proved [1,2,4,12] that
{\bf Lemma VI.} For every $G$-equivariant vector field $ \phi \in \M$,
$ \phi = f(x) \cdot \grad ~,~ \phi : x \to T_x \s [x] $ \EOS
Actually, by the same construction which yields lemma VI, one can also
prove that:
{\bf Lemma VII.} For every $G$-equivariant vector field $ \phi \in \M$, $
\phi : x \to
T_x M^x $ \EOS
Notice that Lemma VII implies Lemma VI. A simplest proof of lemma VII is
obtained by
remarking that if $\~x \simeq x + \eps f(x)$, then by equivariance of $f$,
$g \~x
\simeq gx + \eps f (gx )$, so that $gx = x ~=>~ g \~x = \~x$.
Finally, we should recall some results concerning the geometry of $\~M
\approx \Om$
and the geometry of group action.
Since $\Om$ is a semialgebraic manifold, we can define on it a (Whitney)
stratification in the sense of algebraic geometry [13]; call $E_i^\a$ the
corresponding
strata, and
$$ {\^E}^\a = \cup_i E_i^\a \eqno(21) $$ so that
$$ {\^E}^{\a +1} \sse \pa {\^E}^\a \eqno(22) $$
and the generic stratum will be ${\^E}^0$. It is remarkable that this
stratification
is compatible with the isotropy stratification $\S$, i.e. that each
isotropy stratum is
the union of algebraic strata. This means in particular that nearly all
points in $M$
have the same isotropy type.
The points on strata other than the generic one will have greater isotropy
groups, and
correspondingly smaller $G$-orbits. Indeed, if we consider the generic
stratum $\~M_0 $
in $\~M$, and correspondingly $ M_0 = \th^{-1} \( \~M_0 \) \sse M$, $M_0$ is a
$G$-fiber bundle, $M_0 = \~M_0 \times G$. In general, $M$ is foliated but
not fibered
by $G$, as the leaves (the $G$-orbits) through $x \in M \backslash M_0 $ are not
isomorphic to leaves through $x_0 \in M_0$; the non-generic strata are
therefore also
called {\it singular}.
These considerations also suggest the following construction: let $B \ss M$ be a
subset of $M$ such that $\th (B) = \~M$, i.e. $P \simeq \Om$, or $B$ meets any
$G$-orbit in $M$ in one and only one point.
{\bf Lemma VIII.} If $G$ is a compact Lie group and $M$ a smooth
$G$-manifold, then
there exists a $B_0$ with the above properties and such that $B_0$ is a smooth
submanifold (in general, with boundary) of $M$. \EOS
Once we have find a $B$, clearly $M = G (B)$, and any point $X \in M$ can
be identified by a pair $x = (b,g)$ of coordinates $ b \in B$, $g \in G$,
such that
$$ x = gb \eqno(23) $$
We stress that the rep[resentation (23) is {\it not} unique, in general,
for a given
$B$: e.g., in (23) we could also choose $g' = gh$, with $h \in G_b$.
If the $B$ is chosen as the $B_0$ of lemma VIII, we have that
{\bf Lemma IX.} Let $M = G(B_0 )$, with $B_0$ as in lemma VIII;
consider a smooth curve $x(s) \in M$; then, we can write $x(s) = g(s)
\cdot b(s)$, with $b(s) $ a smooth curve in $B_0$, and $g(s)$ a smooth
curve in $G$.
Therefore, in general, a smooth dynamical system (3) on the $G$-manifold
$M$ can be
also written as
$$ \eqalign{
{\dot g} (t) =& F_1 (g,b) \cr
{\dot b} (t) =& F_2 (g,b) \cr } \eqno(24) $$
As we have remarked before, however, the evolution of the $\th$'s depends
only on the
$\th$'s themselves, so that (24) is actually in "triangular" form
$$ \eqalign{
{\dot g} (t) =& F_1 (g,b) \cr
{\dot b} (t) =& F_2 (b) \cr } \eqno(25) $$
We will summarize our discussion as follows:
{\bf Splitting lemma.} {\it Let $M$ be a smooth $G$-manifold, with $G$ a
compact Lie
group. Let $B$ be a smooth submanifold of $M$, with $\th (B) = \~M \approx
\Om$, where
$\Om = M/G$ is the $G$-orbit space, and $\th : M \to \R^k$ is the map
associated to an
integrity basis. Any point $x \in M$ is then written (in a non-unique way)
as $x=gb$,
where $g \in G$, $b \in B$. Let a smooth $G$-equivariant dynamical system
${\dot x} =
f(x)$ be defined on $M$; this induces a $G$-invariant dynamical system
${\dot \th} =
\~f (\th )$ on $\Om$. Also, with the above representation of $x$, the
dynamical system
is written in the triangular form (25), and is therefore equivalent to (the
semidirect
product of) an autonomous dynamics in $\Om$ (equivalently, in $B$) and a
$\om$-dependent dynamics on $G$.}
It should be stressed that if $G$ is not compact, the splitting lemma does
not hold in
general, but we are however still guaranteed that the splitting is possible
locally.
Results going in the same direction as the splitting lemma can be found in the
literature; see e.g. [14] and [15], to quote two recent works, and also see
next section to see existing results which can be unified by the above
splitting lemma.
It seems, however, that the result has not been stated - and presumably is
not known -
in general terms, as the {\it ad hoc} proofs of these works do also confirm.
%\vfill \eject
\section{4. Discussion I - General features}
Before considering some examples in full detail, we would like to
discuss some special case and applications of the splitting lemma.
We start by general remarks.
{\it Remark 1.} If $f(x) = ( \grad V)(x)$, i.e. if the dynamical
system describes a gradient flow, $f(x)$ is obviously orthogonal to
the $G$-orbit $\om [x]$ at any $x \in M$. Therefore, the dynamics
along the $G$-orbit is absent, and we just have the dynamics in
orbit space.
{\it Remark 2.} The situation considered in Remark 1 was the one
considered by Michel [1-3]. It should be stressed that, although on
the basis of physical considerations he only considered the problem
of critical points for an invariant potential, not only his
construction is general enough to deal with the general case, but
also his original proof does immediately apply to the general case
as well. {\it Our splitting lemma should therefore be seen as a
corollary to Michel theorem} [1].
{\it Remark 3.} A particularly interesting situation arises when
the dynamics in $\Om$ is simple. If $\om_0 \in \Om$ is a fixed point
for this dynamics, the quantities $\th_i$ can be considered as
constants of the motion (for initial data in $\om_0$), or as
asymptotic constants of motion (for initial data in the basis of
attraction of $\om_0$). On the orbit $\om_0$, we can then apply the
momentum map reduction.
{\it Remark 4.} Indeed, this corresponds to saying that the whole
dynamics takes place on $G$ (i.e. on the $G$-orbit $\om_0$).
{\it Remark 5.} In the language of Lie-point symmetries, the
existence of a fixed point for the dynamics in $\Om$ corresponds to
the existence of $k$ simultaneous conditional symmetries [16], or
conditional constants of motion\footnote{$^1$}{It was pointed out to me
separately by G. Saccomandi and by G. Marmo that these were already
known to Amaldi and Levi Civita.} [17], see also Remark 3.
{\it Remark 6.} If $G_x \in C [H]$, the $G$-orbit through $x$ is
isomorphic to $G / G_x \simeq G/H$.
{\it Remark 7.} We have noticed earlier that if $x(0) = x_0$, then
$x (t) \in M^{x_0}$ $\all t \ge 0$. We could therefore apply the
construction - and the splitting lemma - to $M^{x_0}$ rather than to
the whole $M$. Notice that $M^{x_0}$ could, in general, contain
points $y$ such that $G_y$ strictly contains $G_{x_0}$; this is not
the case if $G_{x_0}$ is a maximal isotropy subgroup for the action
of $G$ in $M$. In this case, $M^{x_0}$ is necessarily a fiber
bundle; the fibers correspond to the effective action of $G$ on
$M^{x_0}$. The maximal subgroup of $G$ mapping $M^{x_0}$ into itself
is $N_G (G_{x_0} )$, the normalizer of $G_{x_0}$ in $G$; obviously
$G_{x_0}$ is a normal subgroup of $N_G (G_{x_0} )$, and by
definition it acts as the identity in $M^{x_0}$. Therefore, the
effective action of $G$ in $M^{x_0}$ is given by $D_{x_0} := N_G (
G_{x_0} ) / G_{x_0}$; if $G_{x_0}$ is a maximal isotropy group in
$G$, then the action of $D_{x_0}$ in $M^{x_0}$ is free. (These
remarks are due to Golubitsky and Stewart [12]).
{\it Remark 8.} Suppose that we are interested in knowing if our
system (3) admits periodic orbits. If the dynamics in $\Om$ drives
the system in $M$ to a given $G$-orbit $\om_0$, $\om_0 \simeq G /
G_0$, we would then like to know if the system on this orbit admits
periodic solutions. If there is a periodic (non stationary)
solution $\=x (t) \in \om_0$, then any $ \~x (t) = g \=x (t)$ (with
$g$ fixed) will also be a periodic solution. In other words, $\om_0$
must be fibered by $S^1$ circles (corresponding to the dynamical
system flow). This condition can allow us to assert there are no
periodic solutions, if $\om_0$ does not admit an $S^1$ fibration,
e.g. if it has nonvanishing characteristic classes. Similar
considerations do also apply for quasiperiodic solutions, requiring
a $T^k$ fibration of $\om_0$.
{\it Remark 9.} In some cases, it could be sufficient to have a
local $S^1$ fibration (e.g. if we have local Lie groups rather than
proper ones); the analysis of obstructions to a local $S^1$
fibration of a manifold is considerably more complicate than for
global $S^1$ fibrations; see Gromov {\it et al.} [18].
{\it Remark 10.} Considerations not so different from the above do
also apply for periodic or quasiperiodic solutions to the dynamics
in orbit space. The number of conditional constants of motion would
now be $(k- q)$ for $q$-periodic solutions in $\Om$, and the
coefficients for the dynamics in $G$ would be $q$-periodic functions
of time.
{\it Remark 11.} It is maybe worth mentioning how one could, in
practice, write down the dynamics on $G$. If $\{ L_1 , ... , L_r \}$
are the generators of ${\cal G}$, the Lie algebra of the connected
Lie group $G$, any element $g \in G$ can be written as $ g = {\rm
exp} [\sum_i \lambda_i L_i ]$, $\lambda_i \in \R$, so that the
dynamics in $G$ can be mapped into a dynamics for the $\lambda_i$'s.
{\it Remark 12.} Notice also that the dynamical system (1) induces
a flow which correspond to a one-parameter group $\Phi$ (with Lie
generator $\phi$); in the case of dynamics on only one $G$-orbit,
this $\Phi$ must be a one-parameter subgroup of $G$. Also, since $G$
transforms solutions into solutions, $\Phi$ must be a
normal subgroup of $G$, i.e. $\phi$ must be an ideal of ${\cal G}$.
{\it Remark 13.} The considerations of Remark 12 (and Remarks 8-10
as well) do not apply if $\om_0$ is not a fixed point for the
dynamics in $\Om$. In this case, to a $q$-periodic solution $\om
(t)$ correspond a space phase $T^q \otimes_\to G$ for the dynamics
in $G$ (here $\otimes_\to$ denotes semidirect product).
\vfill \eject
\section{5. Discussion II - Equivariant bifurcation theory}
We would like to present some further remarks, specifically
concerned with the application of our splitting lemma, and more
generally of Michel theory, to equivariant bifurcation theory.
{\it Remark 14.} It is interesting to remark that, nearly
at the same time as Michel and Radicati [3], considerations
resorting to the same approach based on geometry of
symmetry breaking (Lemma VI) were applied to the $SU(3)$
symmetry breaking by Cabibbo and Maiani [5]; their
discussion lacks the mathematical generality of Michel
theorem [1], but curiously their argument is essentially an
anticipation of the "Symmetric Criticality Principle" of
Palais [19]. The latter is (or can be seen as) an extension
of Michel theorem to the case of variational problems
defined by a functional over a space of sections of a fiber
bundle (see also [20] for a similar extension in the case
of gauge theories and for connections with bifurcation
theory).
{\it Remark 15.} Two basic tools in equivariant bifurcation
theory [21,22,12,23] are the "Equivariant Branching Lemma"
(EBL) of Cicogna and Vanderbauwhede [24] (extended to Hopf
case by Golubitsky and Stewart [12]) and the "Reduction
Lemma" (RL) of Golubitsky and Stewart [12]; these can be seen
as corollaries of Michel theorem, although the original
proofs were given with little or no reference to Michel's
work\footnote{$^2$}{It should be stressed that at least some
of the authors working at the developement of equivariant
bifurcation theory, however, were aware not only of the
existence of Michel theorem, but also of its relevance [22].}.
Also, these simpler result do not require the introduction of
the full Michel theory, but just lemmas V, VI, VII above
(about this matter, see also [25,26]). It is interesting that
these useful tools, which are at the core of applications of
equivariant bifurcation theory, fit so naturally in Michel
theory.
{\it Remark 16.} In the present setting, the EBL
corresponds to bifurcation in the orbit space dynamics
(while the dynamics on group variable remains trivial for
stationary bifurcations, and goes around a $G/G_x = U(1)$
subgroup of $G$ in the Hopf case). The RL corresponds to
Lemma VII above, and the reduced dynamics can be seen as
the dynamics on a group (see [23] for concrete examples in
the simplest cases; see also the next section).
{\it Remark 17.} One of the nicest theoretical
accomplishments of equivariant bifurcation theory is the
work of Krupa [14] on bifurcation from a relative
equilibrium (see also [27]). In the present framework,
this corresponds to bifurcation of a fixed point for the
dynamics in the orbit space (see remark 3); if this
undergoes a Hopf bifurcation, we recover the scenario
described by Krupa (see also [26] for an example).
{\it Remark 18.} In a recent work, Chossat and Koenig [15] have
developed a criterion for the existence of periodic branching
solutions, based on the analysis of orbit space. Such a
result can be seen as an extension of Michel analysis
(originally focused on the existence of critical points for
variational problems, or fixed points for the dynamics) to
periodic solutions. This is indeed quite similar in spirit
to the extension by Golubitsky and Stewart [12] of the EBL
[24] to the case of Hopf bifurcations.\footnote{$^3$}{We remark that,
quite curiously, in [15] some results due to Michel are
attributed to Field and Richardson [28], although these
authors correctly attribute them to Michel in their very
interesting and profound works [28].}
{\it Remark 19.} The last few remarks suggest that Michel
theory of symmetry breaking has been generally overlooked,
and that a number of recent inteersting results fit
naturally into it, although obtained in different ways. It
would be not surprising to discover that this theory has
yet a number of interesting and useful results (maybe in
the form of simple but useful corollaries, see the above
remarks) to offer us, such as the splitting lemma presented
here. I hope that the present discussion can also serve to
induce the readers to familiarize with this beautiful
theory and its developements.
\vfill \eject
\section{6. Examples}
Finally, we would like to consider in details some concrete
examples, showing how the splitting lemma applies to simple
dynamical systems.
{\it Example 1.} Let $M = \R^2$, $G=SO(2)$. Then, any
$G$-equivariant dynamical system is written as $$ {\dot x} = f(x)
\equiv \( \a (\rho ) I + \b (\rho ) J \) x $$ where $\rho = (x^2 +
y^2 )/2$, $I$ is the $2 \times 2$ identity matrix, and $J$ is the
generator of $SO(2)$, $$ J = \pmatrix{0&-1\cr 1&0\cr } $$
The $SO(2) $ orbits are indexed by $\rho \ge 0$, and indeed it is
easy to see that $$ {\dot \rho} = ( x \cdot {\dot x}) = \a (\rho )
\cdot \rho \equiv f_\om (\rho ) $$ (notice that $\rho = 0$ is a
fixed point for any $\a , \b$, as stipulated by Michel theorem).
We can therefore pass to polar coordinates $(\rho , \phi )$; in these we have
$$ \eqalign{
{\dot \rho} =& f_\om (\rho ) \equiv \rho \cdot \a (\rho ) \cr
{\dot \phi} =& f_G (\rho , \phi ) \equiv \b ( \rho ) \cr} $$
which realizes the splitting of the dynamics.
{\it Example 2.} Let $M = \R^4$, $G = SU(2)$ acting with generators
$$
H_1 = \pmatrix{0&0&1&0\cr0&0&0&1\cr-1&0&0&0\cr0&-1&0&0\cr} ~~,~~
H_2 = \pmatrix{0&0&0&-1\cr0&0&1&0\cr0&-1&0&0\cr1&0&0&0\cr} ~~,~~
H_3 = \pmatrix{0&-1&0&0\cr1&0&0&0\cr0&0&0&1\cr0&0&-1&0\cr} $$
Then, $G$-equivariant dynamical systems are of the form
$$ {\dot x} = f(x) \equiv \[ \a (\rho ) I + \sum_{i=1}^3 \b_i (\rho ) K_i
\] x $$
where $\rho = (1/2) \sum_{k=1}^4 x_k^2$ and
$$
K_1 = \pmatrix{0&1&0&0\cr-1&0&0&0\cr0&0&0&1\cr0&0&-1&0\cr} ~~,~~
K_2 = \pmatrix{0&0&0&1\cr0&0&1&0\cr0&-1&0&0\cr-1&0&0&0\cr} ~~,~~
K_3 = \pmatrix{0&0&1&0\cr0&0&0&-1\cr-1&0&0&0\cr0&1&0&0\cr} $$
The $SU(2)$ orbits are indexed by $\rho$, and
$$ {\dot \rho} = \a (\rho ) \cdot \rho $$
Again, $x=0$ is a fixed point for any $G$-equivariant $f$.
Notice that for both the present and the previous example, the
evolution on the $G$-orbits which are fixed points for the dynamics in
the orbit space (i.e. such that $\a (\rho )=0$) is {\it linear}; we
have therefore a spontaneous linearization for asymptotic dynamics
[29]. This fact is related to the fact that we have only {\it one}
invariant ($\rho$) for the $G$ action on $M$.
Notice that all orbits - at the exception of $\rho=0$ - belong to the
same stratum, and for $x \not= 0$ we have $G_x = \{ e \}$, $M^x = \R^4
\backslash \{ 0 \}$, $\om [x] \simeq G$.
The group $G = SU(2)$ is isomorphic to the sphere $S^3$, which admits
a fibration by circles $S^1$, the {\it Hopf fibration}. Indeed, unless
$\b_1 (\rho_0 ) = \b_2 (\rho_0 ) = \b_3 (\rho_0 ) = 0$, the flow by
$f$ on the orbits satisfying $\a (\rho_0 ) = 0$ gives raise to
periodic solutions.
{\it Example 3.} Let $M = \R^3$, $G = SO(3)$. here again the orbits
are indexed by $\rho = (1/2) \sum_{i=1}^3 x_i^2$. Now the
$G$-equivariant dynamical systems are of the form
$$ {\dot x} = f(x) \equiv \a (\rho ) x $$
so that the dynamics along $G$ is trivial, and we only have a
nontrivial dynamics in $\Om$,
$$ {\dot \rho} = \a (\rho ) \cdot \rho $$
Thus, $G$-orbits such that $\a (\rho_0 )=0$ correspond to a sphere
$S^2$ of fixed points. This fact is related to the irreducibility of
the standard (defining) representation of $SO(3)$, not only on $\R$
but also on ${\bf C}$.
Notice that examples 1-3 are also examples of the three types of real
irreducible representations, according to Schur lemma over $\R$ [30].
{\it Example 4.} Let $M = \R^2 $ and $G = Z_2^{(1)} \times
Z_2^{(2)}$, generated by
$$ H_1 = \pmatrix{-1 & 0 \cr 0 & 1 \cr } ~~,~~ H_2 = \pmatrix{1 & 0
\cr 0 & -1 \cr }$$
Now we have two $G$-invariants, $\th_1 = x_1^2$ and $\th_2 = x_2^2$;
any $G$-equivariant dynamical system is of the form
$$ {\dot x_i} = f_i (x_1^2 , x_2^2 ) \cdot x_i $$
The orbit space is the quadrant $\{ x_1 \ge 0 ~,~ x_2 \ge 0 \}$; and
the point $x=0$ has $G_x = G$, points $(x_1 , 0 )$ and $(0, x_2)$
(with $x_1 \not= 0$, $x_2 \not= 0$) have, respectively, $G_x =
Z_2^{(2)}$ and $G_x = Z_2^{(1)}$; points $(x_1 , x_2 )$ such that
$x_1 x_2 \not= 0$ have $G_x = \{ e \}$. Notice that $M^{Z_2^{(1)}} =
(0,x_2 )$, $M^{Z_2^{(2)}} = (x_1 ,0 )$; $M^{\{e\} } = M \backslash [
\{ 0 \} \cup M^{Z_2^{(1)}} \cup M^{Z_2^{(2)}} ]$.
Let us consider the time evolution of $\th_1 , \th_2 $:
$$ \eqalign{
{\dot \th_1} =& 2 f_1 (\th_1 , \th_2 ) \cdot \th_1 \cr
{\dot \th_2} =& 2 f_2 (\th_1 , \th_2 ) \cdot \th_2 \cr} $$
Notice that, in agreement with general results [1-4], the strata are
invariant under the flow, so that the axes cannot be crossed, and the
motion will remain confined in a quadrant. This also means that the
evolution on the (discrete) group $G$ is trivial; this is indeed the
only smooth evolution possible on a discrete group.
{\it Example 5.} Let $M = \R^2$ and $G = \R$ acting as dilations;
notice that here we have a noncompact group.
The $G$-equivariant, i.e. scale-invariant, dynamical systems are of
the form
$$ {\dot x} = A x $$
with $A$ a numerical matrix; thus, we have linear dynamical systems.
The $G$-orbits correspond to straight lines $y=mx$, and are indexed
e.g. by $\th = \tan (x_2 / x_1 )$; points along a $G$-orbit are
indexed by $\rho = \vert x \vert^2$. Notice that $x=0$ is a special
$G$-orbit, and also belongs to all the $G$-orbits.
\footnote{$^4$}{The stratification of the orbit space for noncompact
group actions presents some serious problems, and is not always well
defined. The case of noncompact groups with a centralizer satisfying
a compactness condition is discussed in [2, 22]; a more general
discussion of the geometry of non compact group actions is given in
[4]. Similar problems occur for infinite groups actions (here the
very order relation between subgroups is endangered, as an
infinite group can be isomorphic to one of its subgroups); in some
case it is possible to obtain partial results, see [20].}
The difficulty to write down the dynamical systems according to the
splitting lemma - i.e., in this case, in polar coordinates - is
indeed related to this singularity (although $x=0$ is a simple fixed
point for the dynamics). Notice however that locally, in the
neighbourhood of any nonsingular point $x \not= 0$, we can easily
write the dynamical system in polar coordinates, and the equation for
${\dot \th}$ does not depend on $\rho$. The singularity in $x=0$
forbids a global description.
Notice that if we take $G = SO(2) \times \R$ (where $\R$ act as
before), the problem is not present. Indeed, $G$-equivariant
dynamical systems are now of the form
$$ {\dot x} = ( a I + b J ) x $$
with $J$ as in example 2; in polar coordinates, this is just
$$ {\dot \rho } = a \rho ~~;~~ {\dot \th } = b $$
\vfill\eject
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\parskip=0pt
\parindent=0pt
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\bye