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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

260 Z "G 1U G 2 U G3U G

260 Z "G 1U G 2 U G3U G 4, where (3 1 = {(a.b) E ~2la = o}, G 2 = {(a,b) ~ ~. { b --~ 71 a}, G 3- {(a,b)E P. Ib'-~a}, and G 4- {(a,b) c P. {b-~ ~a}. At various places we will assume that g satisfies the genericity condition (a,b)~ Z. Since -I acts as minus identity each maximal isotropy subgroup determines a direction of bifurcation. (3.4) Theorem: (i) Let B(0)-{v c V} Ilvll O. If p and X 0 are chosen sufficiently small and if (a,b)¢ Z then the direction of bifurcation and stability of the bifurcating solutions is as follows: the four lines G divide the plane of parameters in 8 regions, in each region we give the direction uk'left (subcritical), r-right (supercritical)) and the dimension of the unstable manifold in the four regions with a > O: region\A D e d O(2)- O- a>O,b0. -~ rl a

261 denote the generator of the Lie algebra so(2) of SO(2). Since SO(2) does not have a nontrivial fixed point, X acts as an isomorphism. Let u ~ U, u =0 be arbitrary. Consider the orbit D-{eX-lul e ~ SO(2)}. Since ,0 is compact, h~ assumes its maximum at some point UoED. Of course D-{exp(tX)X'lult ~R}. Choose t I such that exp(tlX)X-lu-u o. Use that X and therefore X -1 commute with exp(tX) and differentiate at t-t I . It follows that h(X- 1 exp(tl X)u)- 0, i.e. u 0 c ker (h). (4.2) Lemma= If (a.b)~ Z, then for x and p sufficiently small there are no solutions in Fix(v.~ ) other than those with maximal isotropy. Proofi The points z ~ Fix(z2-) are characterized by zk-(-l)kZ2_k for k--3 ..... 3. Moreo- ver note that e2(v)- Ilvll2v. Therefore in order to find a zero of Xv *acl(v) +be2(v)-0 it is necessary that v and ct(V) are linearly dependent. Using the previous lemma, we may assume that zt-0. Writing ~.z-e2(z) and using zi-0 and Zk-(-1)kz k for k--3 ..... 3 we obtain the system (use the form of the equivariants as given in the appendix) -4~ZoZ2 2 - ~ z 0 ~.ZO= 0 - ~z 2 -Z2 2 Z 3 + ~-- rl ZoZ2Z 3 - -3~lZ2Z22 - ~Z2- 2~Zo2Z2 ~Z 3 - -3rlZ33- 3rlZ22Z3. There are five solutions to this system, which is precisely the number of maximal isotro- py subgroups sitting above Z2-. The discrepancy between the five and the three gro- ups in the poset comes from the fact, that we identified conjugate subgroups in the poset, but here we have to count them individually. This completes the proof. (4.3) Let us indicate the geometry of some of the fixed point subspaces schematically. Fix(O-) Fix(D~) Fix(O-) Fix(D~) Fix(D~) In order to do the computations one needs the equations for these spaces. We use the notation as in Lauterbach [1988]. lemma 3.2.2. We have Fix(O(2)') - { z~ Fix(E 2 -)l z l" z~- z 3 - 0} which is contained in the following two spaces Fix(D;)- {z, Vi= z2-)t z 1" 0}

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