Some Applications of the Swing Lemma

Abstract

We start with an important definition of G. Czédli [18]. For the trajectories $\mathcal{P}\neq \mathcal{Q}$, let $\mathcal{P}\leq _{C}\mathcal{Q}$ if $\mathcal{P}$ is a hat trajectory, $1_{\text{top}(\mathcal{P})} \leq 1_{\text{top}(\mathcal{Q})}$, and $0_{\text{top}(\mathcal{P})}\nleq0_{\text{top}(\mathcal{Q})}$, see Figure 25.1. Czédli defines ≤  _T as the reflexive and transitive closure of ≤  _C . (The notation in G. Czédli [18] is different.) So for a trajectory $\mathcal{P}$, we can define the closure, $\widehat{\mathcal{P}}$, of $\mathcal{P}$: $\mathcal{Q}\in \widehat{\mathcal{P}}$ iff $\mathcal{P}\leq _{C}\mathcal{Q}$ and $\mathcal{Q}\leq _{C}\mathcal{P}$.