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By E. (Edmond) Ramis, C. (Claude) Deschamps, J Odoux

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38), given (<^, $ ) G X x F, the sequence S^{(p,^) converges to (po, $o) when n —>■oo, where $o is the unique fixed point of the contraction $ A{ipQ, $ ) in the complete metric space F. In other words, the sequence of functions 5'"(^, $ ) converges uniformly to the continuous function (^o> ^o) in the open set U when n oo. 70) then ipo is of class in x, and thus it is of class C^. To that effect, consider the functions ipi and given by (pi{t,x) = x and $i(t,a:) = Id. 71) — {s, cpn{s , x)) ^n{s , x) ds .

13), is of class in a neighborhood of {to,xo). P roof. We divide the proof into several steps. Step 1. Construction of a metric space. Given (to)®o) € -D, take constants a,P > 0 such that S := [^0 —o, to + cc] X B{ xq, 2S) C D. 62) Since the set S is compact and / is continuous, we have K ;= max {||/(t, x)|| : (t,x) € 5 } < -l-oo (recall that a continuous function with values in R has a maximum in each compact set). 63) equipped with the distance d{(p,ip) = sup{||(^(t,x) ~'tp{t,x)\\ : (t,x) e U } .

Let M = S^. Given a vector field X : M —>• TM, each vector X(x) is either zero or is tangent to the circle. If X does not take the value zero, then the whole 5^ is an orbit of the equation x' = X(x). If X has exactly p zeros, then there are 2p orbits. 8 8 . Let M = 5^ c and consider a vector field X : M -¥ TM such that for x E other than the North and South poles the vector X (x) is nonzero and horizontal. Then the equation x' = X (x) has critical points at the North and South poles and all other orbits are horizontal circles (obtained from intersecting the sphere with horizontal planes).

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