Nondegenerate false spikes

Definition 6 Consider a solution $(Q, P)$ to Equations (16 )–(17 ). 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.

11.2 Nondegenerate false spikes