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

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

158 1. ~X(z,t) =

158 1. ~X(z,t) = -r(=,t)h(=,t), (z,t) e Wx(V). 2..y(z,t) e C(G,), ~ (,,t) e wx(v). 3. For an, e S,, h(,,t) e S,, t e (t_(,),t+(,)). Let y E U. Then X(y) = d/~(~X(y))l,=o = d/dsC~Cy, s)y)l,=o + dlds(h(y, s))l,=o But d/ds(7(y,s)y)l,=o e Tv and d/ds(h(y,s))lo=O ~. N v. Consequently, these terms are precisely XT(y) and XN(y) respectively. Hence ~xN(y,t) = h(y,t), (y,t) e Wx(U) These arguments reprove the following result due to Krupa: Proposition 6.1 Let X E C~(TM) and a be a ~X-invariant G-orbit. Let U be a G-invariant tubular neighbourhood of a. Then there ez'ists a smooth map 7 : Wx(U) --* G such that: I. ~X(y,t) = 7(y,t)~XN(y,t), (y,t) ~ WX(U). 2. .y(y,t) ~ C(G~), all (y,t) e Wx(~). Definition 6.4 Let X E C~(TM) and a be a ~X.invariant G-orbit. We say a is generic (for X or ~x ) if ~x is normally hyperbolic at a. Just as for the case of diffeomorphisms, we may give equivalent characterizations of genericity. Proposition 6.2 Let X E C~(TM) and a be a ~X-invariant G-orbit. Let S = {S=Iz E a} be a family of slices for ~. Let X = XT -I- XN be the corresponding tangent and normal decomposition of 2(. The following statements are equivalent: 1. a is generic. $. The element 0 E HESS(X, a) has multiplicity dim(a). S. SPEC(XN, a) has precisely dim(a) eigenvalues of real part zero. 4. For any z E a, XNIS= has z as a hyperbolic equilibrium. 7 Models for relative periodic orbits Suppose that X E C~(TM) mad E is a compact ~x. and G-invariant subset of M and there exists z E E, T > 0, such that E = G(~x([0,T])) and E is not a group orbit. Necessarily, E is a smooth G-invariant submanifold of M and the orbit space ~/G is diffeomorphic to S 1. We call ]C a relative periodic orbit of X. Assume T > 0 chosen to be minimal with respect to the property that E = G(~x([0,T])). Let 0 : ~ --* E/G = S 1 denote the orbit map. Then 8 induces a non-zero vector field X* on S 1 characterized by 0~ x = ~x'8. Clearly, S 1 is the unique periodic orbit of ~x" and the period is T.

159 In this section we wish to describe the possible dynamics for equivariant flows on ~, covering a periodic flow on S 1. To do this, we start by classifying G-manifolds which have orbit space S 1. Our first, and main step, is the description of K-principal bundles over S 1 , where K is a compact Lie group. Let K be a compact Lie group and p : E --* S 1 be a principal K-bundle over S 1. For the general theory, background and classification of principal bundles, we refer the reader to [2]. Let Prin(K, S 1) denote the set of isomorphism classes of principal K-bundles over S 1. It is well-known that P~in(K, S ~ ) -~ ~o(a) -~ ~o(a/G °) For reference, we shall give a direct and simple proof of these isomorphisms. Theorem 7.1 There is a natural bijection X : K/K° -* Prin(K, S 1). Proof." We start by defining X : K~ K° -'* Prin(K,S~) • As usual, we let H : K --* K/K ° = P denote the quotient map. Let z E P. Identify S 1 with [0,2r]/(0 = 2~r). Choose ~ E H -1 (z). Let E ¢ be the free K-space defined by Be = [0, 2~] × K/~, (3) where (2~r,k) ~ (0,k(), k E K. Associated to E ¢, there is a natural projection PC : E ¢ --* S 1 and PC : E¢ -'* $1 has the structure of a K-principal bundle over S 1. Suppose (~ E H-~(z). Since ( and (l lie in the same path-component of G, it follows by standard theory that E ¢ and E ¢' are isomorplfic as K-principal bundles over S 1. Hence we may define X(Z) e Prin(K, S 1) by taking X(Z) to be the isomorphism class of the bundle E ¢, for any ~ E H-l(z). The inverse of X is simply constructed by showing that, up to isomorphism, every K-principal bundle over S 1 may be represented in the form (3). We omit the straightforward details. | Notations: Given z E K/K ° = P, let Z(z) denote the cyclic subgroup of P generated by z. Let p(z) denote the order of Z(z). Let Rep(z) = {~ e KtlI(() = z, ¢~z) = e} That is, Rep(z) consists of all representative generators of Z(z) (see Definition 4.3). If E ttep(z), let E ¢ = X(Z), that is E ¢ is the principal K-bundle over S 1 defined by (3). Let pc : E ¢ --* S ~ denote the associated projection map. Suppose that z E P and ~ E Rep(z). Associated to (, we define the K-equivariant flow E¢ on E ¢ by z~(0,k) = (0+ t,k)/~, t e It, k e K, (4) where (2~r, k) ,-~ (0,k(). Clearly, every trajectory of E ¢ is periodic with (prime) period 2p(z)r. We call ~¢ the canonical K-flow on E ¢. Now suppose that H is a closed subgroup of the compact Lie group, G. Before we give our main results, we need to briefly discuss a natural equivalence between N(H)/H-princi- pal bundles over S 1 and G-fiber bundles over S 1 with fiber G/H. Suppose that ~ is a G-manifold, G acts monotypically on ~ with isotropy type (H) and ~/G ~- S 1. Identifying ~/G with S 1, we may regard ~ --* S 1 as a G-fiber bundle over S 1, with fibers G/H.

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