Collisions and spirals of Loewner traces

We analyze Loewner traces driven by functions asymptotic to κ 1 − t . We prove a stability result when κ ≠ 4 , and we show that κ = 4 can lead to nonlocally connected hulls. As a consequence, we obtain a driving term λ ( t ) so that the hulls driven by κ λ ( t ) are generated by a continuous curve for all κ > 0 with κ ≠ 4 , but not when κ = 4 , so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by κ 1 − t and a conceptual proof of the corresponding results of Kager, Nienhuis, and Kadanoff.