5 years ago

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

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

160 Lemma 7.1 Let H be a

160 Lemma 7.1 Let H be a closed subgroup of G. Let FB(G,H,S 1) denote the set of iso- morphism classes of G-fiber bundles over S I with fibers G / H. There is a natural bijeetion A :FB(G,H,S 1) --, Prin(N(H)/H,S 1) Proof: Let E G FB(G,H, S1). Define A(~2) = EH. If we take the obvious free N(H)/H- action on E H, it is clear that A(]C) E Prin(N(H)/H, $1). Conversely, to define the inverse of A, let E be an N(H)/H-principal bundle over S 1. Define A-I(E) to be the fiber product G XN(H) E. Then A-I(E) has the structure of a G-fiber bundle over S ~, fibers GIH. • Lemma 7.2 Let H be a closed subgroup of G. Let E G FB(G,H,S 1) and set A(E) = E H E Prin(N(H)/H, S1). If~ is a G-equivariant flow on E, then is an equivariant flow on E H. Conversely, given any N( H)/ H-equivariant floto q~II on E H there exists a unique G-equivariant flow • on ~ such that ~[~H = ~bH. Similar statements hold for equivariant vector fields and diffeomorphisms. Suppose E E F B( G, H, S 1) corresponds to E E Prin( N ( H) / H, $1). Choose an explicit representative E ¢ for E, where ~ E Rep(z), z E N(H)/N(H) °. Let ~¢ denote the G-fiber bundle G XN(H) E ¢ over S 1. Thus, V.¢ is an explicit representative for E. The canonical flow E ¢ on E ¢ determines a flow on E¢ which we shall also denote by E¢. Each trajectory of =¢ is periodic, period 2p(z)r. Of course, there will in general be many possible choices of ~ yielding the same isomorphism class in FB(G, H, $1). However, the associated flows E ¢ will always be periodic with the same period 2p(z)r. Proposition 7.1 Let H be a closed subgroup of the compact Lie group G. Set K = N(H)/H. Let z G K/K °, ~ E Rep(z). Let • be a G-equivariant flow on ~¢ and assume that the induced flow on S 1 is not trivial. Then: I. For each z G~¢, closure(~x(R)) is isomorphic to a lotus of dimension at most rkCK, ZCz)) + 1. 2. For all O E S 1, elosure(~x(it)) n ~o is isomorphic to T r x Zq, where r < rk(K, Z(z)) and p( )lq. 3. There ezist a smooth G-equivariant map 7 : ~¢ x It --, G and a smooth map p : S 1 x It ---, It such that for all z G ~¢, (a) ~x(t) = 7(z,t)Z¢(z,p(p~x(z),t)), t E It. (b) 7(z,t) E C(Gx), all t E It. 4. If toe define ~'ffi = dosure(~(it)), z E ~¢, then ~ = {~'zlz E ~¢} is a ~-invariant foliation of E ¢. On each leaf Y~z, ~ is either a periodic ~ (dira(~x) = I) or a rational lotus floto. Proof: By virtue of Lemma 7.2, it is no loss of generality to assume that • is a K-eq~fivari- ant flow on the principal K-bundle E ¢. Since both ~' and Z. ¢ are K-equivariant and cover a non-trivial periodic flow on S a, we may (implicitly) define the map p : S ~ × tt --* tt by { = e s x tt p(~,0) =0, aeS 1

161 A straightforward application of the implicit function theorem proves that p is smooth. l~eparametrizing time, using p, it is no loss of generality of assume that p(O, t) = t, (0, t) E SXxR. Fix 0 E S 1, z E E~. Clearly, dosure(~,(R)) N E~ = closure({~(z, 2-~')ln e z} Hence, by Lemma 5.2, closure(~,(R))f3E~ is isomorphic to T" × Zq, where r < rk(K, Z(z)) and p(z)lq, proving 2. Set r = closure(~,(R)) f3 E~ and regard r as a subgroup of K with generator h = @2~(x). Following Smale[*5, page 797], we now suspend ~2~ and consider the manifold A = (r × R)/,.,, where (g,t + 2r) ~ (gh, t), g E r, t e R. The product abelian group structure on r × R drops down to an abelian Lie group structure on A. But A is K-eqttivariantly diffeomorphic to closure(~,(R)) by the map induced from (g,t) ~-* ~(gz,t). In particular, A is connected and is therefore a toms, proving 1. Statement 4 follows by translating the tori closure ~,(R) through E ¢ using K. Finally (3) follows using Lemma 6.2 (alternatively, see Krupa[14, Theorem 2.2]). • Proposition 7.2 Let H be a closed subgroup of G and set K = N(H)/H. Let z E K/K °, E Rep(z) and let PE : E --* S 1 be a G-fiber bundle over S a, fiber G/H. Suppose that E corresponds to E ¢ E Prin(K, S1). Let X be a G-equivariant vector field on ~ t~ith associated flow ~x. Assume that the induced flow on S 1 is non-trivial. Then there e~rist arbitrarily small G-equivariant perturbations X* of X such that: I. For all x e r. , e losu~ ~ 5 ' ( It ) ) is ~omorph e to a lotus o y d ime,~ ian r k ( g , Z(,))+*. Z. For all 0 e S 1, closure~'(It)) N V.o is isomorphic to a Z(z)-maxlmal subgroup of K. Proof: Both statements follow immediately from Lemma 6.5. • To condude this section, we show the relation between Proposition 7.2 and a result of Krupa[14, Theorem 5.1]. Let H be a closed subgroup of G. Let A be a dosed subset of G/H. Define Obviously, G^ is a closed subgroup of G. G^ = {g e GIgA = A} Lemma 7.3 Let A be a closed subset of G/H and suppose Then I. G^ c N(). 2. GA acts transitively on A. 3. G^/H is monogenic. rk( N( GA)IGA ) = rk( N( lt)l H, II(GA/ tt) ) - dlm(A)

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