5 years ago

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

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

198 A straightforward

198 A straightforward calculation shows that the discriminant of L, the caustic, is given by :- ~F ~2F C L = { qa V: H pa W* such that ~--(p,q) = 0 and det 3--~-(p,q) = 0}. If we regard F : W*~V -~ ~ as an unfolding, that is a family of functions on W* parameterised by V, then C L is the set of parameter values q for which F(.,q) has non-Morse critical points - the local bifurcation set of F. Although F is G-invariant, the functions F(.,q) on W* are only invariant under Gq, the isotropy subgroup at q of the action of G on V. If V G denotes the space of fixed points for the action of G on V, then the restriction lqw,evC is a family of G-invariant functions on W*. Moreover, any such family can be extended to a family on W* ~V. It follows that any genetic property of the restricted families can be regarded as a genetic property of the full family. An example of such a property is given by the following result. Proposition 1.2 If V G = {0} then generic G-invariant Morse families F : W*@V ~ ~ satisfy :- 3 2 F det ~(0,0) # 0. 3p2 Hence generic invariant caustics do not pass through isolated fixed points of the action of G on X. Proof By regarding the restricted Morse family lqw.@vG as a family of functions on W* parameterised by V G , we obtain a jet mapping:- 2 j2F1vG : V G ~ JG(W*,IR)0 which takes values in the space of 2-jets at 0 of G-invariant functions on W*. Generically this mapping is transverse to (a stratification of) the subvariety consisting of 2-jets with non-Morse critical points. This subvariety has codimension greater than 0, and so j2Flv o will miss it if V G = {(3}.

Remark 1.3 More generally, the organising centre f--- F(.,0) of a generic G-invariant Morse family must be a germ that can appear in a generic family of G-invariant germs parameterised by V G. If f is 5~G-simple [AGV, §17.3] this means that the codimension of the NG orbit of f in the space of G-invariant germs which vanish at 0 will be less than or equal to the dimension of V G. Moreover the restriction F'tw,~v o will be an ~G versal unfolding of L Here 2G is the group of germs of G-equivariant diffeomorphisms of W*. 199

Reading grade 6 2.A.5.b - mdk12
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