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

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

154 Definition 6.1 ([6])

154 Definition 6.1 ([6]) Let f E Differ(M) and a C M be an f.invariant G orbit. We say a /s generic (for a), if ot is normally hyperbolic for f. Remark 6.1 We recall from [6] that if w is generic for f then there exist smooth stable and unstable manifolds for f through a. Moreover, an equivariant version of Hartman's linearization theorem holds to the effect that in a neighbourhood of a, f is equivariantly topologically conjugate to the normal map Nf : N(a) --* N(~). Our immediate aim is to give spectral characterizations of normal hyperbolicty for invariant G-orbits. Lemma 6.1 Let a be an f-invariant G-orbit. There ezists p >_ 1 such that fP]a is equivariantly isotopic to the identity. Proof: Let x e ~. Choose p > 1 such that fP(z) E (N(G~)/G~) ° and apply Proposi- tion 5.2. • Given an f-invariant G-orbit ~, let a(f) > 1 be the smallest integer > 1 such that fa(1) i~ is equivariantly isotopic to the identity. Definition 6.2 Let f E Differ(M) and ~ C M be an f-invariant G-orbit. An f-admissible pair (U~ U I) of tubular neighbourhoods for ~ consists of a pair of G-invariant tubular neigh- bourhoods of ot satisfying closure(f (V)) C tr', 0 < i < ~(f). Lemma 6.2 Let el be an f-invariant G-orbit. Let (U, U') be an f-admissible pair of tubular neighbourhoods for a and denote the corresponding slice families by $ = {$zlx E a}, $' = {$~lx E ~}, respectively. There exist smooth maps 7 : U .-* G, h : U ---* U' satisfying: 1. fc,(l)(y) = 7(y)h(y), y E U. 2. h : U ~ U ~ is an equivariant embedding. 3. h : Sx ~ S~, all a E a. 4. 7(y) ~ C(G~), aU y e U. Proof: Replacing f by f'~(l) it is no loss of generality to assume a(f) = 1. Fix z E a. By Proposition 5.2, f(z) = cz, for some c e C(Gz). Define ]: ~¢~ ~ M by ](y) --- c-Zf(y). Since e e C(G2), f is G~-equivariant and so ] extends uniquely to a G-equivariant map ] : G- S~ --* M. Note that on U = G. Sx, we have f(y) - ~(y)f(y), where "~ : U ~ G is smooth and ~(y) E C(Gy), all y E U. Indeed, this fonows by observing that for all y E Sx, C(Gu) D C(G~). Now ]la = Ida. Let a : A C G/G~ ~ G be an admissible section of G over G/Gz. Assume first that ](S~) C a(A)S~. Let if" : S~ × A --* a(A)S~ be the map defined by ff'(y,a) = a(a)y. Let r/= (rh,~) = (pa)-z. Define h: S~ --* S~, ~ : Sz -'* G by h(y) = ,~(](y)), y e S~ ~(y) = ~(,~(](y))), y e s.

155 Since ] and ~ are smooth, so are h and 7. By Proposition 5.2, ~(y) E C(G~), all y E S=. Also since ] is an (equlvarlant) diffeomorphism, h : Sz -~ S: is an embedding. Extend 9, h equivariantly to U. Clearly, h : U --, U' is an equlvariant embedding. Further, fa(1)(y) = ~/(y)~/(y)h(y), y E U Define .~(y) = -~(y)?(y), y ~ r:. Since ~(y),~(y) ~ C(G~), it follows that "y(y) ~ C(G~,). Tt rem~ns to prove the c~e when /(S,) ¢ ~(A)S;. Extend ] as a G-equiwiant map to U. Since 5'= is G=-equivariantly contractible, it follows by standard results in the the theory of G-manifolds (see [2, Chapter 6]), that ] is G-equivariantly isotopic, within U I, to a G-equivariant diifeomorphism of U' sending Sx to S~ and fixing x. Hence we may write ]la as a composite fk... fl of G-equivariant embeddings such that :.(S~) C a(A)S~, 1 < i < k. Applying the previous argument to each fiIS~ the result follows I= Remark 6.2 I. The representalon of f~(f) as the composition 7h is analogous to the tangent and normal decomposition studied by Krupa[14]. Similar results, for maps, have been obtained by Chossat and Golubitsky [4]. 2. Suppose that f satisfies the hypotheses of Lemma 6.2 and that in addition (1) f is equivariantly isotopic to the identity and (2) f(y) E U, all y E U. It is reasonable to ask when we can find a map 7 : U --~ G such that f(y) = "y(y)y, y E U with ~/(y) E C(Gu), all y E U. This question has been investigated in [7] where is is shown that we may find a smooth map 7 satisfying these conditions if, for example, all G-orbits in U have the same dimension. If the dimension of G-orbits varies it is in general not possible to find 7 : U -+ G which is smooth or even continuous. For more details and examples we refer the reader to [7]. Let S 1 act on C by multiplication. Let R + denote the set of non-negative real numbers. Clearly, C/S 1 _~ R +. Suppose that A : 1~ n ~ Rn is linear. Denote the spectrum of A by spectrum(A). We recall the following result from [6]. Lemma 6.3 ([6, Theorem G]) Let H be a closed subgroup of G and E be a G-vector bundle over G/H. Suppose A : E ~ E is a G-vector bundle map covering the equivariant diffeomorphism a : G/H -* G/H. Given z E G/H, choose g E G such that ga(z) = x and set spec(A, x,g) = spectrum(gA :Ex ~ Ex)/S 1. Then spec(A, x,g) is independent of the choice of x and g and depends only on A. In the sequel we let spec(A,G/H) denote the common value of spec(A, z, g) given by Lemma 6.3. Remark 6.3 1. We note that spec(A, x, g) is defined as a subset of tt +. If f]a 7k Id and the H-representations on fibers of E are non-trivial, we cannot define the spectrum of A as a subset of C.

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