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

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

164 I. x( P') has

164 I. x( P') has Poineard map P', all P' E Q. ~. x(P')I(M \ U) = X, 1" E O. s. x(P x) = x. Proof: The proof is based on a simple spreading of isotopies argument. For details we refer the reader to [6, Section 6]. I Remark 8.2 If we assume X is C r, 1 < r < vo, rather than smooth in 8.1, then we can only construct a continuous map X : Q "-' C~-1(TM) satisfying the conditions of the Lernma: Contrary to the statement of [6, Section 6,Lemma C], the vector fields x(P ~) that are constructed in the proof are only of class C r-1. Proposition 8.2 Let X E C~(TM) and ~ C M be a relative periodic orbit of X, There ezist arbitrarily C a° small perturbations X n of X such that ~, is a generic relative periodic orbit of X ~. Proof: Let (D, D ~, pX, p) be a Poincar~ system for E. By Lemma 6.5, we may perturb pX to P~ so that ~ 13 D t is a generic invarlant G.orbit of P~. The result follows by Lemma 8.1 and Proposition 8.1. I 8.3 Tangential and normal decomposition Suppose ~] is a relative periodic orbit of X consisting of points of isotropy type (H). Set K = N(H)/H. We may choose z E K/K °, ( E Rep(z) such that ~] and ~]¢ are isomorphic as G-fiber bundles over S 1. As in Section 7, we let E¢ denote the flow on ~¢ induced from the canonical flow on E¢. For the remainder of the section, we shall identify ~] and E¢ and regard E¢ as a flow on ~... Let q~ :/V(E) --* ~ denote the normal bundle of E and fix a tubular map r : N(E) --, M mapping N(~]) equivariantly and diffeomorphically onto an open tubular neighbourhood Uof~. Fix 8 E S 1, z E ~0 H. Let ~ denote the trajectory of ~¢ through z. By construction of the canonical flow, ~ is a periodic orbit, period 2p(z)~. Let q~ : N ~ -* ~ be the H-vector bundle over ~; defined by restricting .N'(~) to ~. Set D ~ = r(N~). We note that D ~ is an embedded H-invariant submanifold of M which is transverse to ~ along D ~ N ~ = ~. Just as in Section 6, we may construct H-eqnlvariant vector fields XT, XN- on D ~ such that 1. XID '~ = Xr + XN.. 2. XT is tangent to G-orbits. 3. XN. is tangent to D ~. The H-eqnivariant vector fields XT, XN. extend to G-equivariant vector fields on U which we denote by XT, XN¢ respectively. We call XT and .~I~N¢ the tangential and normal components of X associated to (. Note that for each g E G, gD ~ is invariant by ~xN..

165 Proposition 8.3 Let P. be a relative periodic orbit of X fi C~(TM) consisting of peints of isotropy type (~). Choose ~ e ~¢(~)/~(H) °, ¢ e Rep(z) such that ~ is isomo~hic to ~¢ and identify Z u~ith ~¢. Let U be a G-invariant tubular neighbourhood of ~ and let X = XT + XN¢ denote the associated tangential and normal decomposition of X. There exist an open neighbourhood Wx(U) of U x {0} in U xR and a smooth map 7 : Wx(U) ~ G such that 1. ~x(z,t) = 7(z,t)~xm(z,t), all (z,t) 6 Wx(U). ~. 7(z,t) e C(G=), all (z,t) 6 Wx(U). 3. ~ is generic for X if and only if it is generic for XN~. ~. ~ is generic for X if and only if t~ is a hyperbolic periodic orbit Of XN¢ID 'c. Proof: Use the arguments of the proof of Proposition 6.1. | References [1] J. F. Adams. Lectures on Lie Groups, (Benjamin, New York, 1969). [2] G. E. Bredon. Introduction to Compact Z~'ansformation Groups, (Pure and Applied Mathematics, 46, Academic Press, New York and London, 1972). [3] T. Br6cker and T. tom Dieck. Representations of Compact Lie Groups, (Graduate Texts in Mathematics, Springer, New York, 1985). [4] P. Chossat and M. Golubitsky. 'Iterates of maps with symmetry', Siam J. of Math. Anal., Vol. 19(6), 1088. [5] M. J. Field. 'Equivariant Dynamical Systems', Bull. Arner. Math. Soc., 76(1970), 1314-1318. [6] M. J. Field. 'Equivariant Dynamical Systems', Trans. Amcr. Math. Soc., 259 (1980),185-205. [7] M. J. Field. 'On the structure of a class of equivariant maps', Bull. Austral Math. Soc., 26(1982), 161-180. [8] M. J. Field. 'Isotopy and Stability of Equivariant Diffeomorphisms', Proc. London Math. Soc.(3), 46(1983), 487-516. [9] M. J. Field. 'Equivariant Dynamics', Contemp. Math, 56(1986), 69-95. [10] M. J. Field, 'Equivariant Bifurcation Theory and Symmetry Breaking', J. Dyn. Diff. Equ., Vol. 1(4), (1989), 369-421. [11] M. G. Golubitsky and D. G. Schaeffer. Sinyularities and Groups in Bifurcation The- ory, Vol. I, (Appl. Math. Sci. 51, Sprlnger-Verlag, New York, 1985). [12] M. G. Golubitsky, D. G. Schaeffer and I. N. Stewart. Singularities and Groups in Bifurcation Theory, Vol. II, (Appl. Math. Sci. 69, Springer-Verlag, New York, 1988).

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