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

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

152 Lemma 5.2 Let f 6

152 Lemma 5.2 Let f 6 Dif~(G/H) correspond to the coset g[H] £ N(H)/H. Then 0j([H]) is naturally isomorphic to the monogenic subgroup of N( H)/ H generated by g[H] and f : 0y([H]) ~ Oy([H]) is a group isomorphism. In particular, OA[H]) -~ T" × Zp for some positive integers r, p. Further, if II(T r × Zp) = K E $(N(H)/H), then r

(a) f(=) = ~(=)x, au x E GIH. (b) f(=) ~ cCG.) °, aU = ~ ClH. 153 Proof: The first assertion is trivial since Diff~G(G/H ) and N(H)/H are isomorphic. The second assertion foUows from Field[8, Lemma D] or directly from Proposition 3.2. | For the remainder of the section we discuss the case of G-equivariant vector fields on G/H. Here the situation is much simpler and we refer the reader to Field[f] and Krupa[14] for details of proofs we omit. Proposition ~.3 Let H be a closed subgroup of G. I. Every G-equivariant ~ector field on G/H is smooth. ~. C~(T(G/H)) ~ L(N(H)/H). s. x/ x E C~(T(G/H)), th~,~ ~ a ~x-i.va,~nt /oZiation ~X = {.~'~[x E G/H} of G/H b~ s-dimensional tori satisf71ing: (a) ~x = dosu~(,x(R)), = ~ GIH. (b) yrx = g~rX, all g E G, x E G/H. (c) • 1 respectively. ~. Given X E C~(T(G/H)), there ezist arbitrarily small perturbations X' of.)( such that the corresponding foliation yrx' of G / H is by tori of dimension rk( N ( H )/ H). Lemma 5.4 Let X E C~(T(G/H)). There e~ts a smooth map 7 : G/H ~ L(G) such that ~. 7(~) ~ LCC(G,)), aU x ~ Cla. Proof: By Lemma 3.4, there exists a smooth map 7 : G/H -, L(G) such that for all = E G/a, ~(=) E L(C(G.)) and The result follows. | X(x) = d/ds(exp(sT(X)))[,=o 6 Equivariant dynamics near invariant group orbits In this section we review the basic definitions of genericity for G-orbits left invariant by an equivariant diffeomorphism or flow. Much of what we describe is covered in greater detail in Field[6, 9] and Krupa[14] and so we often omit proofs, refering the reader to the references. Throughout this section, M will denote a riemannian G-manifold. We assume famil- iarity with the basic theory of normally hyperbolic sets for diffeomorplfisms and flows as described in Hirsch, Pugh and Shub[13]. As in Section 4, we start by considering diffeomorphisms.

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