Nondegenerate true spikes

11.1 Nondegenerate true spikes

The definition of a nondegenerate true spike proceeds by running the construction of a true spike backwards. In other words, we start with a solution $(Q, P )$ such that $1 < v∞ (𝜃0) < 2$ and such that $P τ(τ, 𝜃0) → v∞ (𝜃0)$ . Letting $(Q1, P1) = GEq0,τ0,𝜃0(Q,P )$ , we see, by Equation (41

), that $P1τ(τ,𝜃0) → 1 − v∞(𝜃0) < 0$ . Let $(Q2,P2 ) = Inv(Q1,P1 )$ . Then, due to Proposition 1, $P (τ,𝜃 ) → v (𝜃) − 1 2τ 0 ∞ 0$ and $Q (τ,𝜃 ) → 0 2 0$ . Finally, Proposition 2 applies to $(Q ,P ) 2 2$ so that there are smooth expansions in the neighborhood $I$ of $𝜃0$ . In particular, there is a smooth function $q2$ such that $Q2$ converges to $q2$ with respect to any $Ck$ -norm. Moreover, it is important to note that $q2(𝜃0) = 0$ . We are naturally led to [79

, Definition 1.12, p. 987]:

Definition 5 Consider a solution $(Q, P)$ to Equations (16 )–(17 ). Assume $1 < v (𝜃 ) < 2 ∞ 0$ for some $1 𝜃0 ∈ S$ and that

τli→m∞ Pτ(τ,𝜃0) = v∞ (𝜃0).

Let $(Q2,P2 ) = Inv ∘ GEq ,τ ,𝜃 (Q,P ) 0 0 0$ . By the observations made prior to the definition, $(Q2,P2 )$ has smooth expansions in the neighborhood $I$ of $𝜃 0$ . In particular $Q 2$ converges to a smooth function $q2$ in $I$ and the convergence is exponential in any $k C$ -norm. We call $𝜃0$ a non-degenerate true spike if $∂ 𝜃q2(𝜃0) ⁄= 0$ .

The choice of $q ,τ,𝜃 0 0 0$ is unimportant. Note that nondegenerate true spikes have punctured neighborhoods with normal expansions.