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

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

1 Introduction Local

1 Introduction Local Structure of Equivariant Dynamics Mike Field* Let X be a smooth (that is, C °°) vector field on the differential manifold M and denote the flow of X by ~x. As is well-known, a trajectory of X is compact if and only if it is either an equilibrium point or a periodic orbit. In either of these cases we may give very simple models for dynamics on the trajectory: If x is an equilibrium point of X, we have x(~) = 0, ~x(~,~) = ~, t e tt (1) If 7 is a periodic orbit of X, prime period T, we may reparametrize time and identify 3' with S 1 = [0,T]/(0 = T) to obtain x(o) = 1, o e s 1, ~X(o,t) = o + t, (o,t) c s 1 × It (2) Dynamics in a neighbourhood of a hyperbolic equilibrium or periodic orbit are well- understood. Thus, if Zo is a hyperbolic equilibrium point, there exist smooth stable and unstable manifolds through xo and the local flow of X near zo is topologically conjugate to that of the linearized flow ~ = DX(xo)(z) (tIartman's theorem). If 7 is a periodic orbit, the hyperbolicity of 7 may be described in terms of either the Poincarfi map or the Floquet exponents. If 7 is hyperbolic, we again have stable and unstable manifolds through 7 and a version of Hartman's theorem conjugating the local flow to the flow llnearized in the normal bundle of 7- Suppose now that G is a compact Lie group acting smoothly on M and that X is a G-equivariant vector field on M with associated G-equivariant flow ~x We recall that a group orbit a C M is ca~ed a relative equilibrium of X if a is a ~X-invariant subset of M. If E C M is a compact ~X-invariant. subset of M such that (1) E/G ~ S 1 ; and (2) ~x induces a non-trivlal flow on S 1, we call E a relative periodic orbit of X. Relative equilibria and relative periodic orbits of G-equivariant vector fields respectively correspond to equilibria and periodic orbits of (non-equivariant) vector fields. It is easy to show that if 7 is a maximal trajectory of ~x such that the C-orbit of 7, G- % is compact, then G- 7 is either a relative equilibrium or a relative periodic orbit of X. Our aim in this work is to obtain generalisations of (1), (2) for relative equilibria and periodic orbits and to describe the corresponding generic local theory. We shall do this for both equivariant diffeomorphisms and vector fields. "Research supported by the SERC and the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.

143 Much is already known about the structure of relative equilibria and periodic orbits. For example, in Field[5, 6, 9] there is a fairly complete description of dynamics on relative equilibria as well as some results on relative periodic orbits and group orbits left inval"iant by an equivariant diffeomorphism. However, as Krupa has noted (see [14, Section 5]), the results on relative periodic orbits described in [5, 6, 9] are incomplete. In a recent work [14], Krupa has studied both relative equilibria and relative periodic orbits from the point of view of bifurcation theory. Krupa obtains an important decomposition of equivariant vector fields into (equivariant) tangential and normal components which he uses to analyse bifurcations off relative equilibria. He also obtains sharp results on the dynamics on relative periodic orbits that arise through such bifurcations. As much of the foundational material on relative equilibria and relative periodic orbits is widely scattered in the literature, it seemed worthwhile to provide a reasonably complete coverage of the subject in this paper. Although all our results have natural extensions to parametrized families, we have not included foundational material on G-equlvarlant bifurcation theory. For this, the reader may consult [10], [14] as well as the standard reference texts by Golubitsky, Schaeffer and Stewart[ll],[12]. We now describe the contents of this paper, by section. In Section 2, we cover basic technical preliminaries (mainly slice theory) and notation. Section 3 is largely based on [7, 8]. The main technical result proved is Propositionn 3.3 which gives smooth local H-equivariant sections of a homogeneous space G -4 G/H. This result is repeatedly used in the sequel. In Section 4, we analyse the structure of (monogenic) abel]an subgroups of a compact Lie group. The main result (Proposition 4.1) yields a version of the classical maximal torus theory for non-connected groups. Using this result, we define new integer invariants of a compact Lie group which we subsequently use in our study of relative periodic orbits. In Sections 5 and 6, we study dynamics on and near relative equilibria and group orbits invariant by a diffeomorphism. In Sections 7 and 8 we study dynamics on and near relative periodic orbits. Using the results of Section 4, we show that we may associate a group theoretic integer invariant to every relative periodic orbit. Where applicable, we show that this invariant is equal to the invarlant defined by Krupa[14, Section 5]. We also include a description of the Poincar~ map of a relative periodic orbit and indicate how it may be used in local perturbation theory. It is a pleasure to acknowledge helpful conversations with Marry Goluhitsky and Martin Krupa. 2 Technical Preliminaries and Basic Notations We start by recalling some facts about smooth (that is, C °°) actions of a compact Lie group G on a differential manifold M. We refer the reader to Bredons's text [2], especially Chapters 5 and 6, for further details and proofs. For each x E M, let G. z denote the G-orbit through x and Gx be the isotropy subgroup of G at x. Let (Gx) denote the conjugacy class of Gx in G. We say x, y E M are of the same orbit type if (Gx) = (Gu). Let O(M, G) denote the set of conjugacy classes of isotropy subgroups for the action of G on M. We refer to an element r E O(M, G) as an isotropy type. Given a subgroup H of G we let M g denote the fixed point space of the action of H on M. Let N(H) denote the normaliser of H in G, C(H) denote the centraliser of H in

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