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Additional resources for Dynamical Systems and Bifurcations: Proceedings of a Workshop held in Groningen The Netherlands, April 16–20, 1984

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F l o e r and E. Zehnder M a t h e m a t i s c h e s Institut der R u h r - U n i v e r s i t ~ t Bochum 4630 Bochum (West-Germany) i) Introduction, h i s t o r y and e a r l y results. Our aim is to p r e s e n t some recent results and o p e n q u e s t i o n s c o n c e r n i n g the fixed p o i n t p r o b l e m of symplectic m a p s r e l a t e d to the A r n o l d - c o n j e c t u r e . M o s t o f the t o p o l o g i c a l fixed p o i n t results o r i g i n a t e in the search for p e r i o d i c solutions in celestial mechanics.

On S I x 7 1 and XR ~ g ( z ' r e ) r ~~ take sector S I "x" and to be a t t r a c t i n g If we now for is c a u s e d sectors - y ~3 ) + g(z,x 2 +y 2 ) ( x ~3 restricted XN obtained to C ° - e q u i v a l e n c e about XN - 2 ~ ( x ~ Take the X ~_ V_(_~) whos_e sectors. To say m o r e vector of l-mapping expanding on S I x of + h ( z , r 2 ) ~~. a second such situation YN I S' ' then since PYN is a l s o the time 1-mapping of YR and s i n c e XR and YR are of 5 o j a s i e w i c z - t y p e C°-conjugate, This conjugacy between XN IS coordinates conjugacy The this as as a full sectors (a, the with same having z) only XR and YR are to a C ° - c o n j u g a c y expressed in in c y l i n d r i c a l ~ (~,h(r,z)) where h is the XR and YR" construction conjugacies happens neighbourhood boundary.

R, that in [Du2]. naturally and YN I S' on the conjugacy proven extends in b e t w e e n freedom Exactly was we o b t a i n of one for the cone again permits in a d j a c e n t hyperbolic origin and to c o m b i n e sectors. sectors for of r e v o l u t i o n containing hyperbolic in their 38 In the the vector cone To last of case we distinguish field is attracting vector fields sition in YN XN are and some finite this in normal YN as in As seen ~ and we proven that when form with a same parabolic and C°-conjugate XN on the and YN are kind of decompo- hyperbolic and in that of such will try to two sectors a fortiori X N only investigate a hyperbolic - y ~~ ) + g(z,x 2 +y 2) ( x ~3 Y that that X N and depends on sector if for something a vector ~ + y-~) new field +h(z,x 2 +y 2 ) ~ ~ 0 ~ 5 we X has as class inside I with sectors may a same suppose that kind decomposition of XN has, even using is infinitely near the exactly z-axis the is invarlant in different same invariant cones.

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