5 years ago

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

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

156 2. For all n > 1,

156 2. For all n > 1, spec(An,G/H) = [spec(A,G/tt)]". If a is an f-invariant G-orbit, TI : TaM -* TaM and so Taf induces a G-vector bundle isomorphism Naf of the normal bundle Na = TaM/T~ of a covering flot. Lemma 6.4 Let ~ be an f-invariant G-orbit. The following statements are equivalent: I. tr is generic. 2. 1 ¢ spec(Nof, a). 3. a is generic for fa(l). 4. If x G a and we choose g G N(Gx) such that f(z) = gz, then the spectrum of T=(g-lf) : T=M ~ T~M contains precisely dim(a)-eigenavalues of unit modulus. 5. If we write fa(1) in the form 7h given by Lemma 6.~, then h : Sz --* S~ has z as a hyperbolic fixed point, 2 E a. Proof: The equivalence of (1), (2) and (4) is proved in Field[6]. The equivalence of (1), (3) and (5) follows from Remark 6.3 and [6]. I Lemma 6.5 Let (~ be an f.invariant G-orbit. There exist abitrarily C ~ small perturba- tions f of f such that ~ is a generic f -invariant orbit. Proof: If f is isotopic to the identity, the result follows easily using the characterizations of genericity given by (3) or (4) of Lemma 6.4. Otherwise, write f = (f(Nf)-')Nf = HNf. Then H is isotopic to the identity and we may equlvaxiantly perturb H to H ~ so that ~ is a generic orbit of f' = lttf. We omit the (tedious) details. II We conclude this section by examining the somewhat simpler case of G-orbits left invariant by a vector field (otherwise known as relative equilibria.) Lemma 6.6 Suppose that ct is a ~X_invariant G-orbit. There ezists a smooth equivariant vector ftetd ff supported on a neighbourhood of a such that: 1. ff is everywhere tangent to G-orbits. e. (x - X)la - o. Proof: By Lemma 5.4, there exists a smooth map 7 : a ~ L(G) such that ~x(2) = exp(tT(2))2, t E tt and 7(2) E L(C(Gx)), all 2 E ~. Choose a slice Sx at 2 E a. Let !b : S~ --* tt be a smooth Gx-invaxiant real valued function with compact support which is equal to 1 on a neighbourhood of 2 E S~. For y E Sx define 2(y) = ¢(y)(dlds(exp(sT(2))y)l,=o Since 7(2) E C(Gx) C C(Gu), ff is a Gx-equivaxiant vector field on Sx and so, by Lemma 2.1, .~ extends to a G-equivariant vector field on M. By construction, (X-X)Ia - 0.11 Suppose that c~ is a ~X-invarlant G-orbit and that X - 0 on a. That is, assume c~ is a group orbit of equilibria of X. Since X(x) = 0, z E ~, we may define the Hessian H(X, z)

157 of X at all points z E a. Necessarily H(X, z) is a G=-equivariant linear transformation of T=M, z E c~ and obviously H(X,z) vanishes on T=a, z G a. Further, by G-eqttivaxiance, H(X,z), H(X,y) are similar, all z,y E a. Hence we may define SPEC(X,a) to be the set of eigenvalues of H(X, z), any x E a. Of course, a will be normally hyperbolic for {~x if and only if SPEC(X, a) contains precisely dim(a) eigenvalues with real part zero (These eigenvalues are zero since they are the eigenvalues of the Hessian of XIa ). Before giving the next definition, we note that itt acts on C by translations and that CliR It. Definition 6.8 Let a be a ~X-invariant G-orbit. Let ff be an equivarlant vector field satisfying the conditions of Lemma a.a. We define the (reduced) Hessian HESS(X,~) of X along a by HESS(X, ~) = SPEC(X - ,t, a)litt Lemma 6.7 If a be a ~X-invariant G-orbit, then HESS(X, a) is well-defined, indepen- dent of the choice of ~. Proof: Working in terms of flows, the result is immediate from Lemma 6.4. II Next we review Krupa's decomposition of a vector field near a relative equilibrium into tangent and normal components. All of what we say is covered in greater detail in [14, Sections 2,3] (see also Vanderbauwhede, Krupa and Golubitsky[16]). Our approari is, however, slightly different from these authors as it builds from our previous work on diffeomorphisms. Let ot C M be a G-orbit, S = {S=lz E or} be a family of slices for ~r. Fix z E a. Let X be an equivariant vector field on U = G. S=. Let a : A C G/G= --~ G be an admissible local section. For ea~ z E o¢=, define vector subspaces Tz, Nz C TzM by Tz = Tz(a(A)z) T/ By the G=-invariance property of a, we may define, for all g E G, z E S=, T~z = gT,(a(A).-); N~, = T$ This construction defines orthogonal G-subbundles T and N of TMIU. These bundles are precisely those defined by Krupa{14]. Now let X E C~(TU). We define the smooth G-equivariant vector fields XT, XN on U by: XT = projection of X along T. XN = projection of X along N. We remark that by construction, XT is always tangent to G-orbits. Of course, away from a, XN may not be normal to G-orbits and indeed may even be tangential to G-orbits. For each y E a, z E S v, there exist -oo < t_(z) < ~+(z) < oo such that the trajectory of X]U through z is defined on (t_(z),t+(z)) but on no larger interval. Define Wx(tr) = e v, t e Using Lemma 6.2, we choose smooth maps 7 : Wx(U) --* G, h : Wx(U) --* U satisfying

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