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Xm ). It follows corresponds to at most one point in G<2n,m,τ that 1,γ˜ n μst ≤m,τ T<2n,≤m,τ (f ; x0 , . . ,xm ) 1,γ˜ st st n ≤ 2 μst m,τ {P<2n,m,X n (τ )} μ≤m−1,τ T<2n,≤m−1,τ (f ; x0 , . . , xm ) . The Distortion Lemma then implies that 1,γ˜ n μst ≤m,τ T<2n,≤m,τ (f ; x0 , . . ,xm ) 1,γ˜ n ≤ 6 μst ≤m−1,τ T<2n,≤m−1,τ (f ; x0 , . . , xm ) . ✷ This completes an outline of treatment of part (I) of (12) for the 1-dimensional case. To carry out part (I) of (12) in the N-dimensional case (N > 1) we use the same ideas, but have to overcome additional difficulties.

G. 1. ns ≤ ln n. In this case, the above argument fails. Indeed, let ns = [ln n], n Consider the image of the cone Kαn (ps−1 ) after the last loop Lns Kαn (ps−1 ). It is the cone, whose width angle is of order 1. Taking into account possibility that Dfε,p˜ s−1 rotates a vertical vector by π2 it is certainly not possible to fulfill (20) by a small perturbation. The natural idea is to avoid looking at condition (20) after “short” loops. This leads to combinatorial analysis of type Ns of trajectories.

CBMS Reg. Conf. Ser. Math. 38, Amer. Math. , 1978. , Critical manifolds, travelling waves, and an example from population genetics. J. Math. Biol. 14 (1982), 159–176. , Degenerate critical points, homotopy indices and Morse inequalities. J. Reine Angew. Math. 350 (1984), 1–22. , Computing the homotopy braid index. In preparation, 2005. , Introduction to symplectic field theory. Geom. Func. Anal. Special Volume II (2000), 560–673. , Gradient flows within plane fields. Commun. Math. Helv. 74 (1999), 507–529.