LECTURE 4. THE WEINSTEIN CONJECTURE IN THE TIGHT CASE 91We have <T((-co,co)) C JR since b((-co,co)) c R Let us write
00
<T(z) = L akzk
k=kowith k 0 ;::: 2 where we assume that ako =/. 0. If ko were to be odd then <T lc-Eo,co)
would change its sign near 0. This would imply DV n Dr =/. 0 for some T E ( -E:, 0)
which contradicts our assumption. Hence we may assume that ko is even.Then either b((-co,co)) C [O,oo) or b((-co,co)) c (-oo,O]; let us assume without
loss of generality that b((-co,co)) c [O,oo). Denote by 8Dia the set theoretic
boundary of Dia. Then
0 tj. b(Dti\{O})and -8 tJ. b(8D:CJ for 8 > 0 small. Define a := (sign lc-Eo,co)) <T. Then the
Brouwer degree deg(b,Dia,-8) is well defined and coincides with deg(a,Dia,-8)
because we can remove (z ) by a trivial homotopy argument. Now we continue <T
continuously onto D 00 by reflection. Then
ko deg(DE 0 ,a,O)
deg(DE 0 ,a,-8)
deg(D~, a, -8) + deg(D~, a, -8)
2deg(D~,a, -8)
and therefore deg(Dia,b,-8) = ko/2 if 8 is small enough. Write Ur= (ar,br)
and assume that Ur is defined on Dia c D+ by applying the biholomorphic map
'I/; as before. We know that ar(Dia) C D and that ar ---+ 'I/; in C^00 as T / 0.
Moreover br(z) =f. 0 for all z E Dia and br ---+ 0 in C^00 as T / 0. Moreover
br(-co,co) c (-oo,O) for TE (-c,O). Consider the intersection problem DV nDr
bFigure 14. This figure gives a schematic view of the relative position of the
functions b and bt.