Improvement of the Poincaré-Birkhoff Fixed Point Theorem
Improvement of the Poincaré-Birkhoff Fixed Point Theorem
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By induction there are simple arcs C., 1 < i < k from W to Q (k - i + — ) in intcx except for A and Q(k - i + — ) which intersect only at A, and there are simple arcs C* , 1 < i < k from Q(k - i + — ) to B in intg except for B and Q(k - i + -r) which intersect only at B. Now C, C C*, C C* . . . C, ,C* ,, \b is a sequence of arcs from W to B which are pairwise disioint k-1 k-1 except for W and B. There is exactly one fixed point of f in the in- terior of each of the simple closed curves C u C C*, C C* u C C*, C C* u C C* , C C* u i]i . Hence by the previous argument K. Z. 1C - * Z r\.~".L K." - X K,"~J.. K. - "X Ind^C - Ind c C n C* Ind^C.C* = Ind.C.^C*,, for 1 < i < k - 1 and f fll fii f i+l l+l Ind C C* = End ty . So Ind C = Ind \p . By the same argument f k.— 1 k.— 1 t t i Ind C' = Ind i and so Ind C = Ind C'. D Let G Q = f (x,y) -50- y " 0},;md for each integer i let G. = IL (G_) 1 1 where n is as in lemma 2.11. The following easy lemma will be used to show that a certain arc is simple.
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To quickly assess the difficulty of the text, read a short excerpt:
By induction there are simple arcs C., 1 < i < k from W to Q (k - i + — ) in intcx except for A and Q(k - i + — ) which intersect only at A, and there are simple arcs C* , 1 < i < k from Q(k - i + — ) to B in intg except for B and Q(k - i + -r) which intersect only at B. Now C, C C*, C C* . . . C, ,C* ,, \b is a sequence of arcs from W to B which are pairwise disioint k-1 k-1 except for W and B. There is exactly one fixed point of f in the in- terior of each of the simple closed curves C u C C*, C C* u C C*, C C* u C C* , C C* u i]i . Hence by the previous argument K. Z. 1C - * Z r\.~".L K." - X K,"~J.. K. - "X Ind^C - Ind c C n C* Ind^C.C* = Ind.C.^C*,, for 1 < i < k - 1 and f fll fii f i+l l+l Ind C C* = End ty . So Ind C = Ind \p . By the same argument f k.— 1 k.— 1 t t i Ind C' = Ind i and so Ind C = Ind C'. D Let G Q = f (x,y) -50- y " 0},;md for each integer i let G. = IL (G_) 1 1 where n is as in lemma 2.11. The following easy lemma will be used to show that a certain arc is simple.
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