5 years ago

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

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

162 Proof: Set II(Gh/H)

162 Proof: Set II(Gh/H) = C. Thus C E S(N(H)/It). By Proposition 4.1, there is a C-maximal subgroup K of N(H)/H containing Gh/H. Let ~r : N(H) --* N(H)/H denote the quotient map. Then G^ C r-~K C N(H). Clearly, gG^g -1 = G^, for all g E r-lK. Hence 7r-lK C N(G^). Therefore, N(G^)IG^) ~ r-*K/a^ ~_ (r-~K/H)I(G^t~) Since (Tr-IKIH)I(G^/tI) is a torus of dimension dim(K) - dim(A), it follows that rk(N(G^)/G^) > rk(N(H)/H, C) - dim(A) To prove equality, it is enough to show that r-IK/G^ is a maximal torus in N(GA)/G^. Observe that 1r-IK/G^ C (N(GA)/G^) ° ~ (C(GA)/Z(G^)) ° (Proposition 3.2). Conse- quently, if r-IK/G^ is not maximal, we may choose j e C(G^) such that the group r generated by j and G^ contains r-lK as a proper subgroup. But r/H is abelian and clearly contains K as a proper subgroup, violating the maximality of K. Hence r-IK/G^ is a maximal torus in N(G^)/G^. • Proposition 7.3 (eft [14, Theorem 5.1]) Let H be a closed subgroup of G and set K = N(H). Let z E K/K °, ~ E Rep(z) and let p~ : ~ ~ S I be a G-fiber bundle over S 1, fiber G/H. Suppose that ~ corresponds to E ¢ G Prin(K, S 1). Let t¢ be a periodic orbit of Ffl. Then rk(N(G~)/G,~) = rk(g(H)/H, Z(z)). Proof: The result follows immediately from Lemma 7.3. • 8 Equivariant dynamics near relative periodic orbits In tiffs section we define genericity for relative periodic orbits and show how genericity may be characterized in terms properties of a Poincar~ map. Much of what we do is closely based on Field[6, Sections 4,5,6]. Throughout, we shall suppose that M is a riemannian G-manlfold. Let X E C~(TM) and ~ C M be a relative periodic orbit for X. Definition 8.1 The relative periodic orbit ~ is generic if ~ x is normally hyperbolic at ~. Remark 8.1 Let ~ be a generic relative periodic orbit of X. There exist smooth stable and unstable manifolds for ~. Moreover, an equivariant version of Hartman's linearization theorem holds to the effect that in a neighbourhood of ~, ~x is topolo~cally conjugate to the normal flow N(~ X) : N(~) --* N(~). For further details, we refer the reader to [6, Section 5].

8.1 Poincard maps 163 We review the construction of a Poincard map for a relative periodic orbit (see also [6, Section 4]). Suppose X 6 C~(TM) and E C M is a relative periodic orbit for X. Let r : E --* ~/G = S ~ denote the orbit map. Suppose that the period of the flow induced from ~x on S 1 is T > 0. Let q~ : N(E) --* E denote the normal bundle of E. Fix a tubular map r : N(~) ~ M mapping N(~) equivariaatly and diffeomorphically onto the open tubular neighbourhood U of E. Fix 8 6 S 1 and let a C E denote the G-orbit rr-l(8). Set N,, = N(~)I~, r,, = tiN,, and D = rc,(Nc,). The map r~, : Na --* D C M is a G-equivariant embedding onto the G-invariant snbmaaifold D of M. Note that ~ C D and D meets ~ transversally along a. Since X is tangent to E we may and shall suppose r and D chosen so that X is transverse to D. In particular, X will be non-v~mishing on D. Let N'(E) be a G-invaria~at open disc-subbundle of N(~) and set N" = N'(~)lot, D' = r~(N'). Choose ~ > 0, ~ ,~ T. By continuity of fix, we may suppose N'(E) is chosen to be of sufficiently small radius so that for every V 6 D', there e.xists a unique p(y) 6 [2" - e,T + El such that ~x(~.l,p(y)) 6 D. We now define pX : D' --* D by ex(y) = We call (D,D~,pX,p) a Poinear~ system for E. Exactly as in the construction of the Poincard map of a periodic orbit, one may show that pX : D ~ ~ D is smooth and equivariant and p : D ~ ~ R is smooth and invariant. Similarly, one may prove that pX is independent of choices up to smooth equivariant conjugacy. Proposition 8.1 Let X E C~(TM) and E C M be a relative periodic orbit for X. The following statements are equivalent: 1. E is generic. ~. If (D,D',pX,p) is a Poincar~ system for ~Z, the G-orbit a = D fl E is a generic invariant orbit for pX. Proof: See Field[6, Section 4]. I 8.2 From maps to vector fields Let ~ be a relative periodic orbit of X and (D, D ~, pX, p) be a Poincard system for ~,. We say that an open G-invariant neighbourhood U of E is subordinate to (D, D ~, pX,p) if C U ~x(x,[o,p(~)]) xED I Lemma 8.1 Let ~ be a relative periodic orbit of X and (D,D',pX,p) be a Poincar~ system for E. Let U be an open invariant neighbourhood of ~ subordinate to (D, D', pX, p). There e~ists a tubular neighbourhhod V of E, V C G . D l, an open neighbourhood Q of pX in the subspace of C~(D, D') consisting of maps equal to pX outside V O D ~ and a continuous map X : Q ~ C~(TM) such that:

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