11.2 Nondegenerate false spikes

Let us recall the definition of a nondegenerate false spike, [79Jump To The Next Citation Point, Definition 1.11, p. 986]:

Definition 6 Consider a solution (Q, P) to Equations (16View Equation)–(17View Equation). Assume 0 < v∞ (𝜃0) < 1 for some 𝜃0 ∈ S1 and that

lim Pτ(τ,𝜃0) = − v∞ (𝜃0). τ→ ∞

Let (Q ,P ) = Inv(Q, P ) 1 1. By Proposition 2, we get the conclusion that (Q ,P ) 1 1 has smooth expansions in a neighborhood I of 𝜃0. In particular, Q1 converges to a smooth function q1 in I, and the convergence is exponential in any Ck-norm. By Proposition 1, q1(𝜃0) = 0. We call 𝜃0 a nondegenerate false spike if ∂𝜃q1(𝜃0) ⁄= 0.

Note that nondegenerate false spikes have punctured neighborhoods with normal expansions.

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