The Convergence Behaviour of Sequences of Perturbed Morse Ribbon Trees

Having constructed moduli spaces of perturbed Morse ribbon trees in the previous chapter, we want to investigate sequential compactness properties of these moduli spaces. Our starting point is the consideration of certain sequential compactness results for spaces of perturbed Morse trajectories of the three different types we introduced in Chap. 2. We will show that in all three cases, every sequence in the respective moduli space without a convergent subsequence has a subsequence that converges geometrically against a family of trajectories. The notion of geometric convergence will be made precise in Sects. 5.1 and 5.2.