Pin classes II: Small pin classes

Pin permutations play an important role in the structural study of permutation classes, most notably in relation to simple permutations and well-quasi-ordering, and in enumerative consequences arising from these. In this paper, we continue our study of pin classes, which are permutation classes that comprise all the finite subpermutations contained in an infinite pin permutation. We show that there is a phase transition at $\mu\approx 3.28277$: there are uncountably many different pin classes whose growth rate is equal to $\mu$, yet only countably many below $\mu$. Furthermore, by showing that all pin classes with growth rate less than $\mu$ are essentially defined by pin permutations that possess a periodic structure, we classify the set of growth rates of pin classes up to $\mu$.