By Freddy Dumortier
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Extra info for Germs of Diffeomorphisms in the Plane
Is the uniform limit of the sequence LmZ. Y fields. of equation V~(W),  and vector field Y = y Moreover it is clear that [8) considered In order to show that ~ E F o r Z E V~[W) - ~ sL~ From lemma 3 it follows that the continuous ls a solution as a c o n s e q u e n c e in the space of continuous and hence solves our problem, it would suffice to prove that the sequence (L Ys Z) s converges in the C -topology. A direct proof of this convergence use another more elegant method, Lemma 4 : F o r c > all O, d e p e n d i n g on W(c) integers is possible, inspired from S- s and m, m ~ on s and m, s u c h that 1 +~ if but V i with 0 ~ I then LoX i s ~
K] X is separatrices with separatrices R-invariant], os 1 . . . all Oo, ~ . . . other sectors Ok-1 may be c h o o s e n t o k o% = R O [ o o ] S~+ I . . . S k and be i m a g e s o f S1 . . . S~ and o ~ . . ~% by means o f t h e m a p p i n g s R k [ k E ~ ) , n V = U R k [ ~ ] i s some n e i g h b o u r h o o d o f 0 and t h e R k [ ~ ) have k=l pairwise disjoint interiors, The germ of RK[~] has a boundary c o n s i s t i n g of the germ of K+k RK[o ] U R ~ ]. I,K = RK[Si ] w i t h adjacent and Oi,k = R K [ o .
O K = 0o the a s s o c i a t e d Then 1)} in polar line y of X and for X" and d e n o t e Let S i U mod n K' is a "conic characteristic Let us call [ in b e t w e e n conic of K'. in sectors for X with separatrices. of a f u n d a m e n t a l domain E we may y to be 0 ; in case k = 1 we take any c h a r a c t e r i s t i c choose line of o finite type and call it a o to . In this way E = S,i U ... U S~ 32 [reordering and s i n c e 1, . . k] X is separatrices with separatrices R-invariant], os 1 .
Germs of Diffeomorphisms in the Plane by Freddy Dumortier